Chapter IX

Normal Operators on Hilbert Space

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Section IX.1Spectral Measures and Representations of Abelian C*-Algebras[16/0%]
Problem IX.1.1

Let H\mathcal{H} be a Hilbert space and let {Ai}\{ A_i\} be a net in B(H)\mathcal{B}(\mathcal{H}). Prove the following statements.

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(a)

Ai→AA_i \to A (WOT) if and only if ⟨Aih,k⟩→⟨Ah,k⟩\left\langle A_i h, k \right\rangle \to \left\langle Ah, k \right\rangle for all h,kh, k in H\mathcal{H}.

(b)

If sup⁡i∥Ai∥<∞\sup_i \left\| A_i\right\| < \infty and T\mathcal{T} is a total subset of H\mathcal{H}, then Ai→AA_i \to A (WOT) if and only if ⟨Aih,k⟩→⟨Ah,k⟩\left\langle A_i h, k \right\rangle \to \left\langle Ah, k \right\rangle for all h,kh, k in T\mathcal{T}.

(c)

Ai→AA_i \to A (SOT) if and only if ∥Aih−Ah∥→0\left\| A_i h - Ah\right\| \to 0 for all hh in H\mathcal{H}.

(d)

If sup⁡i∥Ai∥<∞\sup_i \left\| A_i\right\| < \infty and T\mathcal{T} is a total subset of H\mathcal{H}, then Ai→AA_i \to A (SOT) if and only if ∥Aih−Ah∥→0\left\| A_i h - Ah\right\| \to 0 for all hh in T\mathcal{T}.

(e)

If H\mathcal{H} is separable, then the WOT and SOT are metrizable on bounded subsets of B(H)\mathcal{B}(\mathcal{H}).

Problem IX.1.2

Show that ball B(H)\mathcal{B}(\mathcal{H}) is WOT compact.

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Problem IX.1.3

Show that Re⁡B(H)\operatorname {Re} \mathcal{B}(\mathcal{H}) and B(H)+\mathcal{B}(\mathcal{H})_+ are WOT and SOT closed.

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Problem IX.1.4

If L:B(H)→CL: \mathcal{B}(\mathcal{H}) \to \mathbb {C} is a linear functional, show that the following statements are equivalent:

  1. LL is SOT-continuous;

  2. LL is WOT-continuous;

  3. there are vectors h1,…,hn,g1,…,gnh_1, \dots , h_n, g_1, \dots , g_n in H\mathcal{H} such that L(A)=∑j=1n⟨Ahj,gj⟩L(A) = \sum_{j=1}^n \left\langle Ah_j, g_j \right\rangle.

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Problem IX.1.5

Show that a convex subset of B(H)\mathcal{B}(\mathcal{H}) is WOT closed if and only if it is SOT closed.

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Problem IX.1.6

Verify the following statement (Example 1.5). If {En}\{ E_n\} is a sequence of pairwise orthogonal projections on H\mathcal{H}, then ∑1∞En\sum_1^\infty E_n converges (SOT) to the projection of H\mathcal{H} onto ⋁{En(H):n≥1}\bigvee \{ E_n(\mathcal{H}): n \geq 1\}.

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Problem IX.1.7

Verify the statements made in the following three examples.

  • (Example 1.6) Let XX be a compact set, Ω\Omega the Borel subsets of XX, μ\mu a measure on Ω\Omega, and H=L2(μ)\mathcal{H} = L^2(\mu ). For Δ\Delta in Ω\Omega, let E(Δ)=E(\Delta ) = multiplication by χΔ\chi_\Delta, the characteristic function of Δ\Delta. Then EE is a spectral measure for (X,Ω,H)(X, \Omega , \mathcal{H}).

  • (Example 1.7) If EE is a spectral measure for (X,Ω,H)(X, \Omega , \mathcal{H}), the inflation, E(n)E^{(n)}, of EE, defined by E(n)(Δ)=E(Δ)(n)E^{(n)}(\Delta ) = E(\Delta )^{(n)}, is a spectral measure for (X,Ω,H(n))(X, \Omega , \mathcal{H}^{(n)}).

  • (Example 1.8) Let XX be any set, Ω=\Omega = all the subsets of XX, H=\mathcal{H} = any separable Hilbert space, and fix a sequence {xn}\{ x_n\} in XX. If {e1,e2,… }\{ e_1, e_2, \dots \} is some orthonormal basis for H\mathcal{H}, define E(Δ)=E(\Delta ) = the projection onto ⋁{en:xn∈Δ}\bigvee \{ e_n : x_n \in \Delta \}. Then EE is a spectral measure for (X,Ω,H)(X, \Omega , \mathcal{H}).

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Problem IX.1.8

For the spectral measures in (1.6), (1.7), and (1.8), give the corresponding representations.

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Problem IX.1.9

If {Ei}\{ E_i\} is a net of projections and EE is a projection, show that Ei→EE_i \to E (WOT) if and only if Ei→EE_i \to E (SOT).

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Problem IX.1.10

For the representation in (VIII.5.5), find the corresponding spectral measure.

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Problem IX.1.11

In Example VIII.5.4, the representation is not quite covered by Theorem 1.14 since it is a representation of L∞(μ)L^\infty (\mu ) and not C(X)C(X). Nevertheless, this representation is given by a spectral measure defined on Ω\Omega. Find it.

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Problem IX.1.12

Let XX be a compact Hausdorff space and let {xn}\{ x_n\} be a sequence in XX. Let {en}\{ e_n\} be an orthonormal basis for H\mathcal{H} and for each uu in C(X)C(X) define ρ(u)\rho (u) in B(H)\mathcal{B}(\mathcal{H}) by ρ(u)en=u(xn)en\rho (u) e_n = u(x_n) e_n. Show that ρ\rho is a representation and find the corresponding spectral measure.

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Problem IX.1.13

A representation ρ:A→B(H)\rho : \mathcal{A} \to \mathcal{B}(\mathcal{H}) is irreducible if the only projections in B(H)\mathcal{B}(\mathcal{H}) that commute with every ρ(a)\rho (a), aa in A\mathcal{A}, are 00 and 11. Prove that if A\mathcal{A} is abelian and ρ\rho is an irreducible representation of A\mathcal{A}, then dim⁡H=1\dim \mathcal{H} = 1. Find the corresponding spectral measure.

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Problem IX.1.14

Show that a representation ρ:C(X)→B(H)\rho : C(X) \to \mathcal{B}(\mathcal{H}) is injective if and only if E(G)≠0E(G) \neq 0 for every non-empty open set GG, where EE is the corresponding spectral measure.

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Problem IX.1.15

Let {Ai}\{ A_i\} be a net of hermitian operators on H\mathcal{H} and suppose that there is a hermitian operator TT such that Ai≤TA_i \leq T for all ii. If {⟨Aih,h⟩}\{ \left\langle A_i h, h \right\rangle \} is an increasing net in R\mathbb {R} for every hh in H\mathcal{H}, then there is a hermitian operator AA such that Ai→AA_i \to A (WOT).

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Problem IX.1.16

Show that there is a contraction τ:B(H)∗∗→B(H)\tau : \mathcal{B}(\mathcal{H})^{**} \to \mathcal{B}(\mathcal{H}) such that τ(T)=T\tau (T) = T for TT in B(H)\mathcal{B}(\mathcal{H}). If ρ:C(X)→B(H)\rho : C(X) \to \mathcal{B}(\mathcal{H}) is a representation, show that the map ρ~:B(X)→B(H)\tilde\rho : B(X) \to \mathcal{B}(\mathcal{H}) constructed in the proof of Theorem 1.14 (the unique extension of ρ\rho to the bounded Borel functions B(X)B(X), determined by ⟨ρ~(ϕ)g,h⟩=∫ϕ dμg,h\left\langle \tilde\rho (\phi ) g, h \right\rangle = \int \phi \, d\mu_{g,h} for all g,hg, h in H\mathcal{H}, where μg,h∈M(X)\mu_{g,h} \in M(X) is the measure representing u↦⟨ρ(u)g,h⟩u \mapsto \left\langle \rho (u) g, h \right\rangle) is given by ρ~(ϕ)=τ∘ρ∗∗(ϕ)\tilde\rho (\phi ) = \tau \circ \rho^{**}(\phi ).

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Section IX.2The Spectral Theorem[23/0%]
Problem IX.2.1

Let NN be a normal operator on H\mathcal{H} with spectral measure EE. Show that λ∈σp(N)\lambda \in \sigma_p(N) if and only if E({λ})≠0E(\{ \lambda \} ) \neq 0. Moreover, if λ∈σp(N)\lambda \in \sigma_p(N), E({λ})E(\{ \lambda \} ) is the orthogonal projection onto ker⁡(N−λ)\ker (N - \lambda ).

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Problem IX.2.2

Let NN be a normal operator on H\mathcal{H} with spectral measure EE. If Δ\Delta is a clopen subset of σ(N)\sigma (N), show that E(Δ)E(\Delta ) is the Riesz idempotent associated with Δ\Delta.

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Problem IX.2.3

Prove Theorem II.5.1 and its corollaries by using the Spectral Theorem.

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Problem IX.2.4

Prove Theorem II.7.6 and its corollaries by using the Spectral Theorem.

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Problem IX.2.5

Obtain Theorem II.7.11 as a consequence of the following result (Theorem 2.3). If NN is a normal operator on H\mathcal{H} with spectral measure EE and B(σ(N))B(\sigma (N)) is the C∗C^*-algebra of bounded Borel functions on σ(N)\sigma (N), then the map ϕ↦ϕ(N)\phi \mapsto \phi (N) is a representation of the C∗C^*-algebra B(σ(N))B(\sigma (N)). If {ϕi}\{ \phi_i\} is a net in B(σ(N))B(\sigma (N)) such that ∫ϕi dμ→0\int \phi_i \, d\mu \to 0 for every μ\mu in M(σ(N))M(\sigma (N)), then ϕi(N)→0\phi_i(N) \to 0 (WOT). This map is unique in the sense that if τ:B(σ(N))→B(H)\tau : B(\sigma (N)) \to \mathcal{B}(\mathcal{H}) is a representation such that τ(z)=N\tau (z) = N and τ(ϕi)→0\tau (\phi_i) \to 0 (WOT) whenever {ϕi}\{ \phi_i\} is a bounded net in B(σ(N))B(\sigma (N)) such that ∫ϕi dμ→0\int \phi_i \, d\mu \to 0 for every μ\mu in M(σ(N))M(\sigma (N)), then τ(ϕ)=ϕ(N)\tau (\phi ) = \phi (N) for all ϕ\phi in B(σ(N))B(\sigma (N)).

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Problem IX.2.6

Verify the statements in the following example (Example 2.5). If μ\mu is a regular Borel measure on C\mathbb {C} with compact support KK, define NμN_\mu on L2(μ)L^2(\mu ) by Nμf=zfN_\mu f = zf for each ff in L2(μ)L^2(\mu ). It is easy to check that Nμ∗f=zˉfN_\mu^* f = \bar{z} f, and hence NμN_\mu is normal.

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(a)

σ(Nμ)=K=\sigma (N_\mu ) = K = support of μ\mu.

(b)

If, for a bounded Borel function ϕ\phi, we define MϕM_\phi on L2(μ)L^2(\mu ) by Mϕf=ϕfM_\phi f = \phi f, then ϕ(Nμ)=Mϕ\phi (N_\mu ) = M_\phi.

(c)

If EE is the spectral measure for NμN_\mu, then E(Δ)=MχΔE(\Delta ) = M_{\chi_\Delta }.

Problem IX.2.7

Let (X,Ω,μ)(X, \Omega , \mu ) be any σ\sigma-finite measure space and put H=L2(X,Ω,μ)\mathcal{H} = L^2(X, \Omega , \mu ). For ϕ\phi in L∞(μ)≡L∞(X,Ω,μ)L^\infty (\mu ) \equiv L^\infty (X, \Omega , \mu ), define MϕM_\phi on H\mathcal{H} by Mϕf=ϕfM_\phi f = \phi f; then MϕM_\phi is normal with Mϕ∗=MϕˉM_\phi^* = M_{\bar\phi }, and σ(Mϕ)=ess-ran⁡(ϕ)\sigma (M_\phi ) = \operatorname {ess-ran}(\phi ), the essential range of ϕ\phi, defined by

ess-ran⁡(ϕ)≡⋂{cl⁡(ϕ(Δ)):Δ∈Ω and μ(X∖Δ)=0}. \operatorname {ess-ran}(\phi ) \equiv \bigcap \{ \operatorname {cl}(\phi (\Delta )) : \Delta \in \Omega \text{ and } \mu (X \setminus \Delta ) = 0\} .

Verify the following statement (Example 2.6(e)). If EE is the spectral measure for MϕM_\phi [so EE is defined on the Borel subsets of σ(Mϕ)=ess-ran⁡(ϕ)⊆C\sigma (M_\phi ) = \operatorname {ess-ran}(\phi ) \subseteq \mathbb {C}], then for every Borel subset Δ\Delta of σ(Mϕ)\sigma (M_\phi ),

E(Δ)=Mχϕ−1(Δ). E(\Delta ) = M_{\chi _{\phi ^{-1}(\Delta )}}.
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Problem IX.2.8

Let NN be a normal operator on H\mathcal{H} with spectral measure EE. Show that if H\mathcal{H} is separable, there are at most a countable number of points {zn}\{ z_n\} in σ(N)\sigma (N) such that E(zn)≠0E(z_n) \neq 0. By Exercise 1, these are the eigenvalues of NN.

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Problem IX.2.9

Show that a normal operator NN is

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(a)

hermitian if and only if σ(N)⊆R\sigma (N) \subseteq \mathbb {R};

(b)

positive if and only if σ(N)⊆[0,∞)\sigma (N) \subseteq [0, \infty );

(c)

unitary if and only if σ(N)⊆∂D\sigma (N) \subseteq \partial \mathbb {D}.

Problem IX.2.10

Let AA be a hermitian operator with spectral measure EE on a separable space. For each real number tt define a projection P(t)=E(−∞,t)P(t) = E(-\infty , t). Show:

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(a)

P(s)≤P(t)P(s) \leq P(t) for s≤ts \leq t;

(b)

if tn≤tn+1t_n \leq t_{n+1} and tn→tt_n \to t, P(tn)→P(t)P(t_n) \to P(t) (SOT);

(c)

for all but a countable number of points tt, P(tn)→P(t)P(t_n) \to P(t) (SOT) if tn→tt_n \to t;

(d)

for ff in C(σ(A))C(\sigma (A)), f(A)=∫−∞∞f(t) dP(t)f(A) = \int_{-\infty }^\infty f(t) \, dP(t), where this integral is to be defined (by the reader) in the Riemann--Stieltjes sense.

Problem IX.2.11

Let NN be a normal operator on H\mathcal{H} with spectral measure EE. If σp(N)\sigma_p(N) is a Borel set, show that E(σ(N)∖σp(N))=0E(\sigma (N) \setminus \sigma_p(N)) = 0 if and only if NN is diagonalizable; that is, there is a basis for H\mathcal{H} consisting of eigenvectors for NN. If σp(N)\sigma_p(N) is not assumed to be a Borel set, is it still possible to characterize diagonalizable normal operators in a similar way? Give an example of a normal operator NN such that σp(N)\sigma_p(N) is not a Borel set.

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Problem IX.2.12

Let NN be a normal operator on H\mathcal{H}. Show that if N=U∣N∣N = U|N| (∣N∣=(N∗N)1/2|N| = (N^*N)^{1/2}) is the polar decomposition of NN, U=ϕ(N)U = \phi (N) for some Borel function ϕ\phi on σ(N)\sigma (N). Hence U∣N∣=∣N∣UU|N| = |N|U (see Exercise VIII.3.21).

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Problem IX.2.13

Let NN be a normal operator on H\mathcal{H}. Show that N=W∣N∣N = W|N| for some unitary WW that is a function of NN.

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Problem IX.2.14

Prove that if AA is hermitian, exp⁡(iA)\exp (iA) is unitary. Is the converse true?

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Problem IX.2.15

Let NN be a normal operator on H\mathcal{H}. Show that there is a normal operator MM such that M2=NM^2 = N and M=ϕ(N)M = \phi (N) for some Borel function ϕ\phi. How many such normal operators MM are there?

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Problem IX.2.16

Define N:L2(R)→L2(R)N: L^2(\mathbb {R}) \to L^2(\mathbb {R}) by (Nf)(t)=f(t+1)(Nf)(t) = f(t+1). Show that NN is normal and find its spectral decomposition. ((X.6.17) is useful here.)

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Problem IX.2.17

Suppose that N1,…,NdN_1, \dots , N_d are normal operators such that NjNk∗=Nk∗NjN_j N_k^* = N_k^* N_j for 1≤j,k≤d1 \leq j, k \leq d. Show that there is a subset XX of Cd\mathbb {C}^d and a spectral measure EE defined on the Borel subsets of XX such that Nk=∫zk dE(z)N_k = \int z_k \, dE(z) for 1≤k≤d1 \leq k \leq d (zk=z_k = the kkth coordinate function) (see Exercise VIII.2.2).

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Problem IX.2.18

If N1,…,NdN_1, \dots , N_d are as in Exercise 17 and each is compact, show that there is a basis for H\mathcal{H} consisting of eigenvectors for each NkN_k. (This is the simultaneous diagonalization of N1,…,NdN_1, \dots , N_d.)

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Problem IX.2.19

This exercise gives the properties of Hilbert--Schmidt operators (defined below).

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(a)

If {ei}\{ e_i\} and {fj}\{ f_j\} are two orthonormal bases for H\mathcal{H} and A∈B(H)A \in \mathcal{B}(\mathcal{H}), then

∑i∥Aei∥2=∑j∥Afj∥2=∑i∑j∣⟨Aei,fj⟩∣2. \sum _i \left\| Ae_i\right\| ^2 = \sum _j \left\| Af_j\right\| ^2 = \sum _i \sum _j \left| \left\langle Ae_i, f_j \right\rangle \right|^2.
(b)

If A∈B(H)A \in \mathcal{B}(\mathcal{H}) and {ei}\{ e_i\} is a basis for H\mathcal{H}, define

∥A∥2=[∑i∥Aei∥2]1/2. \left\| A\right\| _{2} = \left[ \sum _i \left\| Ae_i\right\| ^2 \right]^{1/2}.

By (a), ∥A∥2\left\| A\right\|_{2} is independent of the basis chosen and hence is well defined. If ∥A∥2<∞\left\| A\right\|_{2} < \infty, AA is called a Hilbert–Schmidt operator. B2=B2(H)\mathcal{B}_2 = \mathcal{B}_2(\mathcal{H}) denotes the set of all Hilbert--Schmidt operators.

(c)

∥A∥≤∥A∥2\left\| A\right\| \leq \left\| A\right\|_{2} for every AA in B(H)\mathcal{B}(\mathcal{H}) and ∥⋅∥2\left\| \cdot \right\|_{2} is a norm on B2\mathcal{B}_2.

(d)

If T∈B=B(H)T \in \mathcal{B} = \mathcal{B}(\mathcal{H}) and A∈B2A \in \mathcal{B}_2, then ∥TA∥2≤∥T∥∥A∥2\left\| TA\right\|_{2} \leq \left\| T\right\| \left\| A\right\|_{2}, ∥A∗∥2=∥A∥2\left\| A^*\right\|_{2} = \left\| A\right\|_{2}, and ∥AT∥2≤∥A∥2∥T∥\left\| AT\right\|_{2} \leq \left\| A\right\|_{2} \left\| T\right\|.

(e)

B2\mathcal{B}_2 is an ideal of B\mathcal{B} that contains B00\mathcal{B}_{00}, the finite-rank operators.

(f)

A∈B2A \in \mathcal{B}_2 if and only if ∣A∣≡(A∗A)1/2∈B2\left|A\right| \equiv (A^*A)^{1/2} \in \mathcal{B}_2; in this case ∥A∥2=∥∣A∣∥2\left\| A\right\|_{2} = \left\| \left|A\right|\right\|_{2}.

(g)

B2⊆B0\mathcal{B}_2 \subseteq \mathcal{B}_0 (the compact operators); moreover, if AA is a compact operator and λ1,λ2,…\lambda_1, \lambda_2, \dots are the eigenvalues of ∣A∣\left|A\right|, each repeated as often as its multiplicity, then A∈B2(H)A \in \mathcal{B}_2(\mathcal{H}) iff ∑n=1∞λn2<∞\sum_{n=1}^\infty \lambda_n^2 < \infty. In this case, ∥A∥2=(∑λn2)1/2\left\| A\right\|_{2} = (\sum \lambda_n^2)^{1/2}.

(h)

If (X,Ω,μ)(X, \Omega , \mu ) is a measure space and k∈L2(μ×μ)k \in L^2(\mu \times \mu ), let K:L2(μ)→L2(μ)K: L^2(\mu ) \to L^2(\mu ) be the integral operator with kernel kk. Then K∈B2(L2(μ))K \in \mathcal{B}_2(L^2(\mu )) and ∥K∥2=∥k∥2\left\| K\right\|_{2} = \left\| k\right\|_{2} (see Proposition II.4.7 and Lemma II.4.8).

(i)

Interpret part (h) for a purely atomic measure space.

More information on B2\mathcal{B}_2 is contained in the next exercise.

Problem IX.2.20

This exercise discusses trace-class operators (defined below) and assumes a knowledge of Exercise 19. B1(H)={AB:A and B∈B2(H)}\mathcal{B}_1(\mathcal{H}) = \{ AB : A \text{ and } B \in \mathcal{B}_2(\mathcal{H})\}. Operators belonging to B1(H)\mathcal{B}_1(\mathcal{H}) are called trace-class operators and B1(H)=B1\mathcal{B}_1(\mathcal{H}) = \mathcal{B}_1 is called the trace class.

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(a)

If A∈B1(H)A \in \mathcal{B}_1(\mathcal{H}) and {ei}\{ e_i\} is a basis, then ∑∣⟨Aei,ei⟩∣<∞\sum \left| \left\langle Ae_i, e_i \right\rangle \right| < \infty. Moreover, the sum ∑⟨Aei,ei⟩\sum \left\langle Ae_i, e_i \right\rangle is independent of the choice of basis. (Hint: If A=C∗BA = C^*B, B,CB, C in B2\mathcal{B}_2, show that ∣⟨Aei,ei⟩∣≡12(∥Bei∥2+∥Cei∥2)\left| \left\langle Ae_i, e_i \right\rangle \right| \equiv \frac{1}{2}(\left\| Be_i\right\|^2 + \left\| Ce_i\right\|^2).)

(b)

If {ei}\{ e_i\} is a basis for H\mathcal{H}, define tr⁡:B1→C\operatorname {tr}: \mathcal{B}_1 \to \mathbb {C} by

tr⁡(A)=∑i⟨Aei,ei⟩. \operatorname {tr}(A) = \sum _i \left\langle Ae_i, e_i \right\rangle .

By (a) the definition of tr⁡(A)\operatorname {tr}(A) does not depend on the choice of a basis; tr⁡(A)\operatorname {tr}(A) is called the trace of AA. If dim⁡H<∞\dim \mathcal{H} < \infty, then tr⁡(A)\operatorname {tr}(A) is precisely the sum of the diagonal terms of any matrix representation of AA.

(c)

If A∈B(H)A \in \mathcal{B}(\mathcal{H}), then the following are equivalent: (1) A∈B1A \in \mathcal{B}_1; (2) ∣A∣=(A∗A)1/2∈B1\left|A\right| = (A^*A)^{1/2} \in \mathcal{B}_1; (3) ∣A∣1/2∈B2\left|A\right|^{1/2} \in \mathcal{B}_2; (4) tr⁡(∣A∣)<∞\operatorname {tr}(\left|A\right|) < \infty.

(d)

If A∈B1A \in \mathcal{B}_1 and T∈BT \in \mathcal{B}, then ATAT and TATA are in B1\mathcal{B}_1 and tr⁡(AT)=tr⁡(TA)\operatorname {tr}(AT) = \operatorname {tr}(TA). Moreover, tr⁡:B1→C\operatorname {tr}: \mathcal{B}_1 \to \mathbb {C} is a positive linear functional such that if A∈B1A \in \mathcal{B}_1, A≥0A \geq 0, and tr⁡(A)=0\operatorname {tr}(A) = 0, then A=0A = 0.

(e)

If A∈B1A \in \mathcal{B}_1, define ∥A∥1≡tr⁡(∣A∣)\left\| A\right\|_{1} \equiv \operatorname {tr}(\left|A\right|). If A∈B1A \in \mathcal{B}_1 and T∈BT \in \mathcal{B}, show that ∣tr⁡(TA)∣≤∥T∥∥A∥1\left|\operatorname {tr}(TA)\right| \leq \left\| T\right\| \left\| A\right\|_{1}.

(f)

∥A∥1=∥A∗∥1\left\| A\right\|_{1} = \left\| A^*\right\|_{1} if A∈B1A \in \mathcal{B}_1.

(g)

If T∈BT \in \mathcal{B} and A∈B1A \in \mathcal{B}_1, then ∥TA∥1≤∥T∥∥A∥1\left\| TA\right\|_{1} \leq \left\| T\right\| \left\| A\right\|_{1} and ∥AT∥1≤∥T∥∥A∥1\left\| AT\right\|_{1} \leq \left\| T\right\| \left\| A\right\|_{1}.

(h)

∥⋅∥1\left\| \cdot \right\|_{1} is a norm on B1\mathcal{B}_1. It is called the trace norm.

(i)

B1\mathcal{B}_1 is an ideal in B(H)\mathcal{B}(\mathcal{H}) that contains B00\mathcal{B}_{00}.

(j)

If A∈B1A \in \mathcal{B}_1 and {ei}\{ e_i\} and {fi}\{ f_i\} are two bases for H\mathcal{H}, then ∑i∣⟨Aei,fi⟩∣≤∥A∥1\sum_i \left| \left\langle Ae_i, f_i \right\rangle \right| \leq \left\| A\right\|_{1}.

(k)

B1⊆B0\mathcal{B}_1 \subseteq \mathcal{B}_0. Also, if A∈B0A \in \mathcal{B}_0 and λ1,λ2,…\lambda_1, \lambda_2, \dots are the eigenvalues of ∣A∣\left|A\right|, each repeated as often as its multiplicity, then A∈B1A \in \mathcal{B}_1 if and only if ∑n=1∞λn<∞\sum_{n=1}^\infty \lambda_n < \infty. In this case, ∥A∥1=∑n=1∞λn\left\| A\right\|_{1} = \sum_{n=1}^\infty \lambda_n.

(l)

If AA and B∈B2B \in \mathcal{B}_2, define (A,B)=tr⁡(B∗A)(A, B) = \operatorname {tr}(B^*A). Then (⋅,⋅)(\cdot , \cdot ) is an inner product on B2\mathcal{B}_2, ∥⋅∥2\left\| \cdot \right\|_{2} is the norm defined by this inner product, and B2\mathcal{B}_2 is ∥⋅∥2\left\| \cdot \right\|_{2} complete. In other words, B2\mathcal{B}_2 is a Hilbert space.

(m)

(B1,∥⋅∥1)(\mathcal{B}_1, \left\| \cdot \right\|_{1}) is a Banach space.

(n)

B00\mathcal{B}_{00} is dense in both B1\mathcal{B}_1 and B2\mathcal{B}_2. (For more on these matters, see Ringrose [1971] and Schatten [1960].)

Problem IX.2.21

This exercise assumes a knowledge of Exercise 20. If g,h∈Hg, h \in \mathcal{H}, let g⊗hg \otimes h denote the rank-one operator defined by (g⊗h)(f)=⟨f,h⟩g(g \otimes h)(f) = \left\langle f, h \right\rangle g.

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(a)

If g,h∈Hg, h \in \mathcal{H} and A∈B(H)A \in \mathcal{B}(\mathcal{H}), tr⁡(A(g⊗h))=⟨Ag,h⟩\operatorname {tr}(A(g \otimes h)) = \left\langle Ag, h \right\rangle.

(b)

If T∈B1T \in \mathcal{B}_1, then ∥T∥1=sup⁡{∣tr⁡(CT)∣:C∈B0,∥C∥≤1}\left\| T\right\|_{1} = \sup \{ \left|\operatorname {tr}(CT)\right| : C \in \mathcal{B}_0, \left\| C\right\| \leq 1\}.

(c)

If T∈B1T \in \mathcal{B}_1, define LT:B0→CL_T: \mathcal{B}_0 \to \mathbb {C} by LT(C)=tr⁡(TC)L_T(C) = \operatorname {tr}(TC) (=tr⁡(CT)= \operatorname {tr}(CT)). Show that the map T↦LTT \mapsto L_T is an isometric isomorphism of B1\mathcal{B}_1 onto B0∗\mathcal{B}_0^*.

(d)

If B∈BB \in \mathcal{B}, define FB:B1→CF_B: \mathcal{B}_1 \to \mathbb {C} by FB(T)=tr⁡(BT)F_B(T) = \operatorname {tr}(BT). Show that B↦FBB \mapsto F_B is an isometric isomorphism of B\mathcal{B} onto B1∗\mathcal{B}_1^*.

(e)

If L∈B∗L \in \mathcal{B}^*, show that L=L0+L1L = L_0 + L_1 where L0,L1∈B∗L_0, L_1 \in \mathcal{B}^*, L1(B)=tr⁡(BT)L_1(B) = \operatorname {tr}(BT) for some TT in B1\mathcal{B}_1, and L0(C)=0L_0(C) = 0 for every compact operator CC. Show that ∥L∥=∥L0∥+∥L1∥\left\| L\right\| = \left\| L_0\right\| + \left\| L_1\right\| and that L0L_0 and L1L_1 are unique. Give necessary and sufficient conditions on H\mathcal{H} for B(H)\mathcal{B}(\mathcal{H}) to be a reflexive Banach space.

Problem IX.2.22

Prove that if UU is any unitary operator on H\mathcal{H}, then there is a continuous function u:[0,1]→B(H)u: [0, 1] \to \mathcal{B}(\mathcal{H}) such that u(t)u(t) is unitary for all tt, u(0)=Uu(0) = U, and u(1)=1u(1) = 1.

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Problem IX.2.23

If NN is normal, show that there is a sequence of invertible normal operators that converges to NN.

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Section IX.3Star-Cyclic Normal Operators[8/0%]
Problem IX.3.1

If μ\mu is a compactly supported measure on C\mathbb {C} and f∈L2(μ)f \in L^2(\mu ), ff is a star-cyclic vector for NμN_\mu if and only if μ({x:f(x)=0})=0\mu (\{ x : f(x) = 0\} ) = 0.

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Problem IX.3.2

Prove the following proposition (Proposition 3.2).

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(a)

A vector e0e_0 is a star-cyclic vector for AA if and only if H=cl⁡{Te0:T∈C∗(A)}\mathcal{H} = \operatorname {cl}\{ Te_0 : T \in C^*(A)\}, where C∗(A)=C^*(A) = the C∗C^*-algebra generated by AA.

(b)

A vector e0e_0 is a cyclic vector for AA if and only if H=cl⁡{p(A)e0:p=a polynomial}\mathcal{H} = \operatorname {cl}\{ p(A)e_0 : p = \text{a polynomial}\}.

Problem IX.3.3

If μ1\mu_1 and μ2\mu_2 are compactly supported measures on C\mathbb {C}, show that the following statements are equivalent.

  1. μ1\mu_1 and μ2\mu_2 are boundedly mutually absolutely continuous.

  2. There is an isomorphism V:L2(μ1)→L2(μ2)V: L^2(\mu_1) \to L^2(\mu_2) such that VNμ1V−1=Nμ2VN_{\mu_1}V^{-1} = N_{\mu_2} and VL∞(μ1)=L∞(μ2)VL^\infty (\mu_1) = L^\infty (\mu_2).

  3. There is a bounded bijection R:L2(μ1)→L2(μ2)R: L^2(\mu_1) \to L^2(\mu_2) such that Rp(z,zˉ)=p(z,zˉ)Rp(z, \bar{z}) = p(z, \bar{z}) for every polynomial in zz and zˉ\bar{z}.

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Problem IX.3.4

Show that if NN is a star-cyclic normal operator and λ∈σp(N)\lambda \in \sigma_p(N), then dim⁡ker⁡(N−λ)=1\dim \ker (N - \lambda ) = 1.

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Problem IX.3.5

If NN is diagonalizable and star-cyclic and if σp(N)={λ1,λ2,… }\sigma_p(N) = \{ \lambda_1, \lambda_2, \dots \}, show that NN is unitarily equivalent to NμN_\mu, where μ=∑n=1∞2−nδλn\mu = \sum_{n=1}^\infty 2^{-n} \delta_{\lambda_n} (see Exercise 2.11).

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Problem IX.3.6

Let NN be a diagonalizable normal operator. Show that N≅MN \cong M if and only if MM is a diagonalizable normal operator, σp(N)=σp(M)\sigma_p(N) = \sigma_p(M), and dim⁡ker⁡(N−λ)=dim⁡ker⁡(M−λ)\dim \ker (N - \lambda ) = \dim \ker (M - \lambda ) for all λ\lambda. (Compare this with Theorem II.8.3.)

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Problem IX.3.7

Let UU be the bilateral shift on l2(Z)l^2(\mathbb {Z}). If e0e_0 is the vector in l2(Z)l^2(\mathbb {Z}) that has 11 in the zeroth place and zeros elsewhere, then e0e_0 is a star-cyclic vector for UU. If μ\mu is the compactly supported measure on C\mathbb {C} and V:l2(Z)→L2(μ)V: l^2(\mathbb {Z}) \to L^2(\mu ) is the isomorphism such that Ve0=1Ve_0 = 1 and VUV−1=NμVUV^{-1} = N_\mu, then

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(a)

μ=m=\mu = m = normalized arc length on ∂D\partial \mathbb {D};

(b)

V−1=V^{-1} = the Fourier transform on L2(m)=L2(∂D)L^2(m) = L^2(\partial \mathbb {D}).

Problem IX.3.8

Suppose N1,…,NdN_1, \dots , N_d are normal operators such that NjNk∗=Nk∗NjN_j N_k^* = N_k^* N_j for 1≤j,k≤d1 \leq j, k \leq d and suppose there is a vector e0e_0 in H\mathcal{H} such that H\mathcal{H} is the only subspace of H\mathcal{H} containing e0e_0 that reduces each of the operators N1,…,NdN_1, \dots , N_d. Show that there is a compactly supported measure μ\mu on Cd\mathbb {C}^d and an isomorphism V:H→L2(μ)V: \mathcal{H} \to L^2(\mu ) such that VNkV−1f=zkfVN_k V^{-1} f = z_k f for ff in L2(μ)L^2(\mu ) and 1≤k≤d1 \leq k \leq d (zk=z_k = the kkth coordinate function) (see Exercise 2.17).

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Section IX.4Some Applications of the Spectral Theorem[10/0%]
Problem IX.4.1

If NN is a normal operator show that ran⁡N\operatorname {ran} N is closed if and only if 00 is not a limit point of σ(N)\sigma (N).

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Problem IX.4.2

Give an example of a non-normal operator AA such that 00 is an isolated point of σ(A)\sigma (A) and ran⁡A\operatorname {ran} A is closed. Give an example of a non-normal operator BB such that ran⁡B\operatorname {ran} B is closed and 00 is not an isolated point of σ(B)\sigma (B).

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Problem IX.4.3

If H\mathcal{H} is a nonseparable Hilbert space find an example of a nontrivial closed ideal of B(H)\mathcal{B}(\mathcal{H}) that is different from B0(H)\mathcal{B}_0(\mathcal{H}).

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Problem IX.4.4

Let (X,Ω,μ)(X, \Omega , \mu ) be the measure space obtained in the proof of the following theorem (Theorem 4.6: if NN is a normal operator on H\mathcal{H}, then there is a measure space (X,Ω,μ)(X, \Omega , \mu ) and a function ϕ\phi in L∞(X,Ω,μ)L^\infty (X, \Omega , \mu ) such that NN is unitarily equivalent to MϕM_\phi on L2(X,Ω,μ)L^2(X, \Omega , \mu )) and show that L1(X,Ω,μ)∗L^1(X, \Omega , \mu )^* is isometrically isomorphic to L∞(X,Ω,μ)L^\infty (X, \Omega , \mu ).

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Problem IX.4.5

Show that H\mathcal{H} is separable if and only if every collection of pairwise orthogonal projections in B(H)\mathcal{B}(\mathcal{H}) is countable.

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Problem IX.4.6

If (X,Ω,μ)(X, \Omega , \mu ) is a measure space, then (X,Ω,μ)(X, \Omega , \mu ) is σ\sigma-finite or L2(μ)L^2(\mu ) is finite dimensional if and only if every collection of pairwise orthogonal projections in {Mϕ∈L∞(μ)}\{ M_\phi \in L^\infty (\mu )\} is countable.

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Problem IX.4.7

If N=∫z dE(z)N = \int z \, dE(z) and ε>0\varepsilon > 0, show that ran⁡E({z:∣z∣>ε})⊆ran⁡N\operatorname {ran} E(\{ z : \left|z\right| > \varepsilon \} ) \subseteq \operatorname {ran} N.

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Problem IX.4.8

(Calkin) Let M\mathcal{M} be a linear manifold in H\mathcal{H} and show that M\mathcal{M} has the property that M\mathcal{M} contains no closed infinite dimensional subspaces if and only if whenever A∈B(H)A \in \mathcal{B}(\mathcal{H}) and ran⁡A⊆M\operatorname {ran} A \subseteq \mathcal{M}, then AA is compact.

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Problem IX.4.9

Show that the extreme points of {A∈B(H):0≤A≤1}\{ A \in \mathcal{B}(\mathcal{H}) : 0 \leq A \leq 1\} are the projections.

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Problem IX.4.10

(Halmos) If NN is a normal operator, show that there is a hermitian operator AA and a continuous function ff such that N=f(A)N = f(A). (Hint: Use the following result, Theorem 4.6. If NN is a normal operator on H\mathcal{H}, then there is a measure space (X,Ω,μ)(X, \Omega , \mu ) and a function ϕ\phi in L∞(X,Ω,μ)L^\infty (X, \Omega , \mu ) such that NN is unitarily equivalent to MϕM_\phi on L2(X,Ω,μ)L^2(X, \Omega , \mu ).)

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Section IX.5Topologies on B(H)\mathcal{B}(\mathcal{H})[6/0%]
Problem IX.5.1

Let A\mathcal{A} be a subalgebra of B(H)\mathcal{B}(\mathcal{H}) containing 11. Recall that for 1≤n≤∞1 \leq n \leq \infty, H(n)\mathcal{H}^{(n)} denotes the direct sum of H\mathcal{H} with itself nn times, and if A∈B(H)A \in \mathcal{B}(\mathcal{H}), A(n)A^{(n)} is the operator on H(n)\mathcal{H}^{(n)} defined by A(n)(h1,…,hn)=(Ah1,…,Ahn)A^{(n)}(h_1, \dots , h_n) = (Ah_1, \dots , Ah_n); if S⊆B(H)\mathcal{S} \subseteq \mathcal{B}(\mathcal{H}), S(n)≡{A(n):A∈S}\mathcal{S}^{(n)} \equiv \{ A^{(n)} : A \in \mathcal{S}\}. Proposition 5.3 states that the SOT closure of A\mathcal{A} is

{B∈B(H):for every finite n, Lat⁡A(n)⊆Lat⁡B(n)}. \{ B \in \mathcal{B}(\mathcal{H}) : \text{for every finite } n, \ \operatorname {Lat} \mathcal{A}^{(n)} \subseteq \operatorname {Lat} B^{(n)}\} .

Show that if B∈SOT-cl⁡AB \in \text{SOT-}\operatorname {cl} \mathcal{A}, then BB belongs to the set defined above.

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Problem IX.5.2

Show that B00\mathcal{B}_{00} is SOT dense in B\mathcal{B}.

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Problem IX.5.3

If {Ak}\{ A_k\} and {Bk}\{ B_k\} are sequences in B(H)\mathcal{B}(\mathcal{H}) such that Ak→AA_k \to A (WOT) and Bk→BB_k \to B (SOT), then AkBk→ABA_k B_k \to AB (WOT).

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Problem IX.5.4

With the notation of Exercise 3, show that if Ak→AA_k \to A (SOT), then AkBk→ABA_k B_k \to AB (SOT).

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Problem IX.5.5

Let SS be the unilateral shift on l2(N)l^2(\mathbb {N}) (II.2.10). Examine the sequences {Sk}\{ S^k\} and {S∗k}\{ S^{*k}\} and their relation to Exercises 3 and 4.

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Problem IX.5.6

(Halmos) Fix an orthonormal basis {en:n≥1}\{ e_n : n \geq 1\} for H\mathcal{H}.

?
(a)

Show that 0∈0 \in weak closure of {nen:n≥1}\{ \sqrt{n} e_n : n \geq 1\} (Halmos [1982], Solution 28).

(b)

Let {ni}\{ n_i\} be a net of integers such that nieni→0\sqrt{n_i} e_{n_i} \to 0 weakly. Define Aif=ni⟨f,eni⟩eniA_i f = \sqrt{n_i} \left\langle f, e_{n_i} \right\rangle e_{n_i} for ff in H\mathcal{H}. Show that Ai→0A_i \to 0 (SOT) but {Ai2}\{ A_i^2\} does not converge to 00 (SOT).

Section IX.6Commuting Operators[12/0%]
Problem IX.6.1

If S⊆B(H)\mathcal{S} \subseteq \mathcal{B}(\mathcal{H}), show that S′=S′′′\mathcal{S}' = \mathcal{S}'''.

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Problem IX.6.2

If S⊆B(H)\mathcal{S} \subseteq \mathcal{B}(\mathcal{H}), show that S′\mathcal{S}' is always a SOT closed subalgebra of B(H)\mathcal{B}(\mathcal{H}).

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Problem IX.6.3

Prove the following proposition (Proposition 6.1).

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(a)

If A=A1⊕A2⊕⋯A = A_1 \oplus A_2 \oplus \cdots is a bounded operator on H=H1⊕H2⊕⋯\mathcal{H} = \mathcal{H}_1 \oplus \mathcal{H}_2 \oplus \cdots and B=[Bij]∈B(H)B = [B_{ij}] \in \mathcal{B}(\mathcal{H}), then AB=BAAB = BA if and only if BijAj=AiBijB_{ij} A_j = A_i B_{ij} for all i,ji, j.

(b)

If B=[Bij]∈B(H(n))B = [B_{ij}] \in \mathcal{B}(\mathcal{H}^{(n)}), BA(n)=A(n)BBA^{(n)} = A^{(n)} B if and only if BijA=ABijB_{ij} A = A B_{ij} for all i,ji, j.

Problem IX.6.4

Let H\mathcal{H} be a Hilbert space of dimension α\alpha and define S:H(∞)→H(∞)S: \mathcal{H}^{(\infty )} \to \mathcal{H}^{(\infty )} by S(h1,h2,… )=(0,h1,h2,… )S(h_1, h_2, \dots ) = (0, h_1, h_2, \dots ). SS is called the unilateral shift of multiplicity α\alpha.

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(a)

Show that A=[Aij]∈{S}′A = [A_{ij}] \in \{ S\} ' if and only if Aij=0A_{ij} = 0 for j>ij > i and Aij=Ai+1,j+1A_{ij} = A_{i+1, j+1} for i≥ji \geq j.

(b)

Show that A=[Aij]∈{S}′′A = [A_{ij}] \in \{ S\} '' if and only if Aij=0A_{ij} = 0 for j>ij > i and Aij=Ai+1,j+1=A_{ij} = A_{i+1, j+1} = a multiple of the identity for i≥ji \geq j.

Problem IX.6.5

What is {Nμ⊕Nμ}′\{ N_\mu \oplus N_\mu \} '? {Nμ⊕Nμ}′′\{ N_\mu \oplus N_\mu \} ''?

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Problem IX.6.6

If A\mathcal{A} is a subalgebra of B(H)\mathcal{B}(\mathcal{H}), show that A\mathcal{A} is a maximal abelian subalgebra of B(H)\mathcal{B}(\mathcal{H}) if and only if A=A′\mathcal{A} = \mathcal{A}'.

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Problem IX.6.7

Find a non-normal operator that is similar to a normal operator. (Hint: Try dim⁡H=2\dim \mathcal{H} = 2.)

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Problem IX.6.8

Let μ\mu be a compactly supported measure on C\mathbb {C} and let X\mathcal{X} be a separable Hilbert space. A function f:C→Xf: \mathbb {C} \to \mathcal{X} is a Borel function if f−1(G)f^{-1}(G) is a Borel set when GG is weakly open in X\mathcal{X}. Define L2(μ,X)L^2(\mu , \mathcal{X}) to be the equivalence classes of Borel functions f:C→Xf: \mathbb {C} \to \mathcal{X} such that ∫∥f(x)∥2 dμ(x)<∞\int \left\| f(x)\right\|^2 \, d\mu (x) < \infty. Define ⟨f,g⟩=∫⟨f(x),g(x)⟩ dμ(x)\left\langle f, g \right\rangle = \int \left\langle f(x), g(x) \right\rangle \, d\mu (x) for ff and gg in L2(μ,X)L^2(\mu , \mathcal{X}).

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(a)

Show that L2(μ,X)L^2(\mu , \mathcal{X}) is a Hilbert space. Define NN on L2(μ,X)L^2(\mu , \mathcal{X}) by (Nf)(z)=zf(z)(Nf)(z) = zf(z).

(b)

Show that NN is a normal operator and σ(N)=\sigma (N) = support μ\mu. Calculate N∗N^*.

(c)

Show that N≅Nμ(α)N \cong N_\mu^{(\alpha )}, where α=dim⁡X\alpha = \dim \mathcal{X}.

(d)

Find {N}′\{ N\} '. (Hint: Use (6.1).)

(e)

Find {N}′′\{ N\} ''.

Problem IX.6.9

Let H\mathcal{H} be separable with basis {en}\{ e_n\}. Let AA be the diagonal operator on H\mathcal{H} given by Aen=λnenAe_n = \lambda_n e_n, where sup⁡n∣λn∣<∞\sup_n \left|\lambda_n\right| < \infty. Determine {A}′\{ A\} ' and {A}′′\{ A\} ''. Give necessary and sufficient conditions on {λn}\{ \lambda_n\} such that {A}′={A}′′\{ A\} ' = \{ A\} ''.

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Problem IX.6.10

Let A\mathcal{A} be a C∗C^*-subalgebra of B(H)\mathcal{B}(\mathcal{H}) but do not assume that A\mathcal{A} contains the identity operator. Let M=⋁{ran⁡A:A∈A}\mathcal{M} = \bigvee \{ \operatorname {ran} A : A \in \mathcal{A}\} and let P=P = the projection of H\mathcal{H} onto M\mathcal{M}. Show that SOT-cl⁡A=A′′P=PA′′\text{SOT-}\operatorname {cl} \mathcal{A} = \mathcal{A}'' P = P \mathcal{A}''.

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Problem IX.6.11

Formulate and prove a polar decomposition for operators between different Hilbert spaces.

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Problem IX.6.12

Let (X,Ω,μ)(X, \Omega , \mu ) be an arbitrary measure space and let L∈L1(μ)∗L \in L^1(\mu )^*.

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(a)

Show that for every ff in L2(μ)L^2(\mu ), there is an hh in L2(μ)L^2(\mu ) such that L(gfˉ)=∫ghˉ dμL(g\bar{f}) = \int g \bar{h} \, d\mu for all gg in L2(μ)L^2(\mu ).

(b)

If f∈L2(μ)f \in L^2(\mu ), let TfTf be the function hh in L2(μ)L^2(\mu ) obtained in part (a). Show that T:L2(μ)→L2(μ)T: L^2(\mu ) \to L^2(\mu ) defines a bounded linear operator and TT commutes with MϕM_\phi for every ϕ\phi in L∞(μ)L^\infty (\mu ).

(c)

In light of parts (a) and (b), compare Theorem 6.6 and Example 20.17 in Hewitt and Stromberg [1975].

Section IX.7Abelian von Neumann Algebras[16/0%]
Problem IX.7.1

Prove the following proposition (Proposition 7.3).

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(a)

If A1,A2,…\mathcal{A}_1, \mathcal{A}_2, \dots are von Neumann algebras, then so is A1⊕A2⊕⋯\mathcal{A}_1 \oplus \mathcal{A}_2 \oplus \cdots.

(b)

If A\mathcal{A} is a von Neumann algebra and 1≤n≤∞1 \leq n \leq \infty, then A(n)\mathcal{A}^{(n)} is a von Neumann algebra.

Problem IX.7.2

Prove the following proposition (Proposition 7.4). Let Aj\mathcal{A}_j be a von Neumann algebra on Hj\mathcal{H}_j, j=1,2j = 1, 2. If U:H1→H2U: \mathcal{H}_1 \to \mathcal{H}_2 is an isomorphism such that UA1U−1=A2U \mathcal{A}_1 U^{-1} = \mathcal{A}_2, then UA1′U−1=A2′U \mathcal{A}_1' U^{-1} = \mathcal{A}_2'.

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Problem IX.7.3

Why are Aμ\mathcal{A}_\mu and Aμ(2)\mathcal{A}_\mu^{(2)} not spatially isomorphic?

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Problem IX.7.4

Show that if XX is any compact metric space, there is a separable Hilbert space H\mathcal{H} and a ∗*-monomorphism τ:C(X)→B(H)\tau : C(X) \to \mathcal{B}(\mathcal{H}). Find the spectral measure for τ\tau.

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Problem IX.7.5

Let A\mathcal{A} be an abelian C∗C^*-subalgebra of B(H)\mathcal{B}(\mathcal{H}) such that A′\mathcal{A}' contains no uncountable collection of pairwise orthogonal projections. Show that the following statements are equivalent:

  1. A\mathcal{A} is a maximal abelian von Neumann algebra;

  2. A=A′\mathcal{A} = \mathcal{A}';

  3. A\mathcal{A} has a cyclic vector and is SOT closed;

  4. there is a finite measure space (X,Ω,μ)(X, \Omega , \mu ) and an isomorphism U:L2(μ)→HU: L^2(\mu ) \to \mathcal{H} such that UAμU−1=AU \mathcal{A}_\mu U^{-1} = \mathcal{A}.

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Problem IX.7.6

Let {Pn}\{ P_n\} be a sequence of commuting projections in B(H)\mathcal{B}(\mathcal{H}) and put A=∑n=1∞3−n(2Pn−1)A = \sum_{n=1}^\infty 3^{-n}(2P_n - 1). Show that C∗(A)C^*(A) is the C∗C^*-algebra generated by {Pn}\{ P_n\}. (Do you see a connection between AA and the Cantor--Lebesgue function?)

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Problem IX.7.7

If A\mathcal{A} is an abelian von Neumann algebra on a separable Hilbert space H\mathcal{H}, show that there is a hermitian operator AA such that A\mathcal{A} equals the smallest von Neumann algebra containing AA. (Hint: Let {Pn}\{ P_n\} be a countable WOT dense subset of the set of projections in A\mathcal{A} and use Exercise 6. This proof is due to Rickart [1960], pp. 293--294. Also see Jenkins [1972].)

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Problem IX.7.8

If XX is a compact space, show that C(X)C(X) is generated as a C∗C^*-algebra by its characteristic functions if and only if XX is totally disconnected. If AA is as in Exercise 6, show that σ(A)\sigma (A) is totally disconnected.

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Problem IX.7.9

If XX and YY are compact spaces and τ:C(X)→C(Y)\tau : C(X) \to C(Y) is a homomorphism with τ(1)=1\tau (1) = 1, show that there is a continuous function ϕ:Y→X\phi : Y \to X such that τ(u)=u∘ϕ\tau (u) = u \circ \phi for every uu in C(X)C(X). Show that τ\tau is injective if and only if ϕ\phi is surjective, and, in this case, τ\tau is an isometry. Show that τ\tau is surjective if and only if ϕ\phi is injective.

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Problem IX.7.10

Let XX and ZZ be compact spaces, Y=X×ZY = X \times Z, and let ϕ:Y→X\phi : Y \to X be the projection onto the first coordinate. Define τ:C(X)→C(Y)\tau : C(X) \to C(Y) by τ(u)=u∘ϕ\tau (u) = u \circ \phi. Describe the range of τ\tau.

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Problem IX.7.11

Adopt the notation of Exercise 9. Define an equivalence relation ∼\sim on YY by saying y1∼y2y_1 \sim y_2 if and only if ϕ(y1)=ϕ(y2)\phi (y_1) = \phi (y_2). Let q:Y→Y/∼q: Y \to Y/{\sim } be the natural map and q∗:C(Y/∼)→C(Y)q^*: C(Y/{\sim }) \to C(Y) the induced homomorphism. Show that there is a ∗*-epimorphism ρ:C(X)→C(Y/∼)\rho : C(X) \to C(Y/{\sim }) such that the diagram

C(X)→ τ C(Y)ρ↓↑q∗C(Y/∼) \begin{array}{ccc} C(X) & \xrightarrow {\ \tau \ } & C(Y) \\ {\scriptstyle \rho } \downarrow & & \uparrow {\scriptstyle q^*} \\ & C(Y/{\sim }) & \end{array}

commutes. Find the corresponding injection Y/∼→XY/{\sim } \to X.

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Problem IX.7.12

If XX is any compact metric space, show that there is a totally disconnected compact metric space YY and a continuous surjection ϕ:Y→X\phi : Y \to X. (Hint: Start by embedding C(X)C(X) into B(H)\mathcal{B}(\mathcal{H}) and use Exercises 7 and 8.)

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Problem IX.7.13

Show that every totally disconnected compact metric space is the continuous image of the Cantor ternary set. (Do this directly; do not try to use C∗C^*-algebras.) Combine this with Exercise 12 to get that every compact metric space is the continuous image of the Cantor set.

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Problem IX.7.14

If A\mathcal{A} is an abelian von Neumann algebra and XX is its maximal ideal space, then XX is a Stonean space; that is, if UU is open in XX, then cl⁡U\operatorname {cl} U is open in XX.

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Problem IX.7.15

This exercise assumes Exercise 2.21, where it was proved that B1∗=B\mathcal{B}_1^* = \mathcal{B}. When referring to the weak* topology on B=B(H)\mathcal{B} = \mathcal{B}(\mathcal{H}), we mean the topology B\mathcal{B} has as the Banach dual of B1\mathcal{B}_1.

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(a)

Show that on bounded subsets of B\mathcal{B} the weak* topology == WOT.

(b)

Show that a C∗C^*-subalgebra of B(H)\mathcal{B}(\mathcal{H}) is a von Neumann algebra if and only if A\mathcal{A} is weak* closed.

(c)

Show that WOT and the weak* topology agree on abelian von Neumann algebras (see Takeda [1951] and Pallu de la Barriere [1954]).

(d)

Give an example of a weak* closed subspace of B\mathcal{B} that is not WOT closed. (See von Neumann [1936].)

Problem IX.7.16

Prove the converse of the following proposition (Proposition 7.6: if e0e_0 is a cyclic vector for A\mathcal{A}, then e0e_0 is a separating vector for A′\mathcal{A}').

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Section IX.8The Functional Calculus for Normal Operators: The Conclusion of the Saga[15/0%]
Problem IX.8.1

What is a scalar-valued spectral measure for a diagonalizable normal operator?

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Problem IX.8.2

Let N1N_1 and N2N_2 be normal operators with scalar-valued spectral measures μ1\mu_1 and μ2\mu_2. What is a scalar spectral measure for N1⊕N2N_1 \oplus N_2?

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Problem IX.8.3

Let {en}\{ e_n\} be an orthonormal basis for H\mathcal{H} and put μ(Δ)=∑n=1∞2−n∥E(Δ)en∥2\mu (\Delta ) = \sum_{n=1}^\infty 2^{-n} \left\| E(\Delta ) e_n\right\|^2. Show that μ\mu is a scalar-valued spectral measure for NN.

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Problem IX.8.4

Give an example of a normal operator on a nonseparable space which has no scalar-valued spectral measure.

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Problem IX.8.5

Prove that the map ρ\rho in the following theorem (Theorem 8.10: if NN is a normal operator on the separable Hilbert space H\mathcal{H} and μ\mu is a scalar-valued spectral measure for NN, then there is a well-defined map ρ:L∞(μ)→W∗(N)\rho : L^\infty (\mu ) \to W^*(N) given by the formula ρ(ϕ)=ϕ(N)\rho (\phi ) = \phi (N) such that (a) ρ\rho is a ∗*-isomorphism and an isometry; (b) ρ:(L∞(μ),weak*)→(W∗(N),WOT)\rho : (L^\infty (\mu ), \text{weak*}) \to (W^*(N), \text{WOT}) is a homeomorphism) is an isometry without using (VIII.4.8).

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Problem IX.8.6

Prove the following proposition (Proposition 8.12). Let NN be a normal operator on a separable space with scalar-valued spectral measure μ\mu, and let ϕ∈L∞(μ)\phi \in L^\infty (\mu ) (so by the Spectral Mapping Theorem, σ(ϕ(N))\sigma (\phi (N)) is the μ\mu-essential range of ϕ\phi). If N=∫z dEN = \int z \, dE, then μ∘ϕ−1\mu \circ \phi^{-1} is a scalar-valued spectral measure for ϕ(N)\phi (N) and E∘ϕ−1E \circ \phi^{-1} is its spectral measure.

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Problem IX.8.7

Show that if μ\mu and ν\nu are compactly supported measures on C\mathbb {C}, the following statements are equivalent:

  1. Nμ⊕NνN_\mu \oplus N_\nu is ∗*-cyclic;

  2. W∗(Nμ⊕Nν)=W∗(Nμ)⊕W∗(Nν)W^*(N_\mu \oplus N_\nu ) = W^*(N_\mu ) \oplus W^*(N_\nu );

  3. μ⊥ν\mu \perp \nu.

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Problem IX.8.8

If MM and NN are normal operators with scalar-valued spectral measures μ\mu and ν\nu, respectively, show that the following are equivalent:

  1. W∗(M⊕N)=W∗(M)⊕W∗(N)W^*(M \oplus N) = W^*(M) \oplus W^*(N);

  2. {M⊕N}′={M}′⊕{N}′\{ M \oplus N\} ' = \{ M\} ' \oplus \{ N\} ';

  3. there is no operator AA such that MA=ANMA = AN other than A=0A = 0;

  4. μ⊥ν\mu \perp \nu.

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Problem IX.8.9

If MM and NN are normal operators, show that C∗(M⊕N)=C∗(M)⊕C∗(N)C^*(M \oplus N) = C^*(M) \oplus C^*(N) if and only if σ(M)∩σ(N)=∅\sigma (M) \cap \sigma (N) = \varnothing.

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Problem IX.8.10

Give an example of two normal operators MM and NN such that W∗(M⊕N)=W∗(M)⊕W∗(N)W^*(M \oplus N) = W^*(M) \oplus W^*(N) but C∗(M⊕N)≠C∗(M)⊕C∗(N)C^*(M \oplus N) \neq C^*(M) \oplus C^*(N). In fact, find MM and NN such that W∗(M⊕N)W^*(M \oplus N) splits, but σ(M)=σ(N)\sigma (M) = \sigma (N).

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Problem IX.8.11

If UU is the bilateral shift and VV is any unitary operator, show that W∗(U⊕V)=W∗(U)⊕W∗(V)W^*(U \oplus V) = W^*(U) \oplus W^*(V) if and only if VV has a spectral measure that is singular to arc length on ∂D\partial \mathbb {D}. (See Exercise 3.7.)

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Problem IX.8.12

If A\mathcal{A} is an abelian von Neumann algebra on a separable space, show that there is a compactly supported measure μ\mu on R\mathbb {R} such that A\mathcal{A} is ∗*-isomorphic to L∞(μ)L^\infty (\mu ). (Hint: Use Exercise 7.7.)

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Problem IX.8.13

(This exercise assumes a knowledge of Exercise 2.21.) Let N=∫z dE(z)N = \int z \, dE(z) be a normal operator with scalar-valued spectral measure μ\mu and define α:B1(H)→L1(μ)\alpha : \mathcal{B}_1(\mathcal{H}) \to L^1(\mu ) by α(T)(Δ)=tr⁡(TE(Δ))\alpha (T)(\Delta ) = \operatorname {tr}(TE(\Delta )). Show that α\alpha is a surjective contraction. What is α∗\alpha^*? [L1(μ)L^1(\mu ) is identified, via the Radon--Nikodym Theorem, with the set of complex-valued measures that are absolutely continuous with respect to μ\mu.]

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Problem IX.8.14

Let π:B(H)→B(H)/B0(H)\pi : \mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H})/\mathcal{B}_0(\mathcal{H}) be the natural map and let A∈B(H)A \in \mathcal{B}(\mathcal{H}).

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(a)

If π(A)\pi (A) is hermitian, show that there is a hermitian operator BB such that A−BA - B is compact.

(b)

If π(A)\pi (A) is positive, show that there is a positive operator BB such that A−BA - B is compact. (See Exercise XI.3.14.)

Problem IX.8.15

(L.G. Brown) If π:B(H)→B(H)/B0(H)\pi : \mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H})/\mathcal{B}_0(\mathcal{H}) is the natural map and AA and BB are hermitian operators such that π(A)≤0≤π(B)\pi (A) \leq 0 \leq \pi (B), then there is a hermitian compact operator KK such that A≤K≤BA \leq K \leq B.

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Section IX.9Invariant Subspaces for Normal Operators[7/0%]
Problem IX.9.1

(Ando) Use the following result (Corollary 9.6: if NN is a normal operator and μ\mu is a scalar-valued spectral measure for NN, then NN is reductive if and only if P∞(μ)=L∞(μ)P^\infty (\mu ) = L^\infty (\mu )) to show that every compact normal operator is reductive.

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Problem IX.9.2

Determine all of the invariant subspaces of a compact normal operator.

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Problem IX.9.3

(Ando) Show that every reductive compact operator is normal. (Also see Rosenthal [1968].)

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Problem IX.9.4

Show that an operator AA is reductive if and only if Lat⁡A=Lat⁡A∗\operatorname {Lat} A = \operatorname {Lat} A^*.

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Problem IX.9.5

Let NN be a normal operator on a Hilbert space H\mathcal{H}, let R\mathcal{R} be an invariant subspace for NN, and let S=N∣RS = N | \mathcal{R}. Show that SS is normal if and only if R\mathcal{R} is a reducing subspace for NN.

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Problem IX.9.6

Let μ\mu be a compactly supported regular Borel measure on C\mathbb {C} and let P2(μ)P^2(\mu ) denote the closure in L2(μ)L^2(\mu ) of the analytic polynomials; that is, P2(μ)P^2(\mu ) is the closed linear span of {zn:n≥0}\{ z^n : n \geq 0\}. Show that P2(μ)P^2(\mu ) is invariant for NμN_\mu and that Nμ∣P2(μ)N_\mu | P^2(\mu ) is normal if and only if P2(μ)=L2(μ)P^2(\mu ) = L^2(\mu ).

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Problem IX.9.7

With the notation of Exercise 6, show that if R\mathcal{R} is a reducing subspace for NμN_\mu and P2(μ)⊆RP^2(\mu ) \subseteq \mathcal{R}, then R=L2(μ)\mathcal{R} = L^2(\mu ).

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Section IX.10Multiplicity Theory for Normal Operators: A Complete Set of Unitary Invariants[14/0%]
Problem IX.10.1

Let AA and BB be operators on H\mathcal{H} and K\mathcal{K}, respectively. Let H0\mathcal{H}_0 and K0\mathcal{K}_0 be reducing subspaces for AA and BB and suppose that A≅B∣K0A \cong B | \mathcal{K}_0 and B≅A∣H0B \cong A | \mathcal{H}_0. Show that A≅BA \cong B.

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Problem IX.10.2

Let μ1,μ2,…\mu_1, \mu_2, \dots be compactly supported measures on C\mathbb {C} such that μn+1≪μn\mu_{n+1} \ll \mu_n for all nn. Show that if MM is any normal operator whose spectral measure is absolutely continuous with respect to each μn\mu_n, then

Nμ1⊕Nμ2⊕⋯≅(Nμ1⊕Nμ2⊕⋯ )⊕M. N_{\mu _1} \oplus N_{\mu _2} \oplus \cdots \cong (N_{\mu _1} \oplus N_{\mu _2} \oplus \cdots ) \oplus M.
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Problem IX.10.3

If μ\mu is Lebesgue measure on [0,1][0, 1], show that Nμ≅NμpN_\mu \cong N_\mu^p for 0<p<∞0 < p < \infty.

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Problem IX.10.4

Let μ\mu be Lebesgue measure on [0,1][0, 1] and characterize the functions ϕ\phi in L∞(μ)L^\infty (\mu ) such that Nμ≅ϕ(Nμ)N_\mu \cong \phi (N_\mu ).

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Problem IX.10.5

Let μ\mu be area measure on D\mathbb {D} and show that NμN_\mu and Nμ2N_\mu^2 are not unitarily equivalent.

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Problem IX.10.6

Let μ\mu be Lebesgue measure on [0,1][0, 1] and let ν\nu be Lebesgue measure on [−1,1][-1, 1]. Show that Nν2≅Nμ⊕NμN_\nu^2 \cong N_\mu \oplus N_\mu. How about Nν3N_\nu^3?

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Problem IX.10.7

Let μ\mu be Lebesgue measure on R\mathbb {R} and N=N = multiplication by sin⁡x\sin x on L2(μ)L^2(\mu ). Find the decompositions of NN obtained in Theorem 10.1 and Theorem 10.16, stated below.

Theorem 10.1: (a) If NN is a normal operator, then there is a sequence (possibly finite) of measures {μn}\{ \mu_n\} on C\mathbb {C} such that μn+1≪μn\mu_{n+1} \ll \mu_n for all nn and N≅Nμ1⊕Nμ2⊕⋯N \cong N_{\mu_1} \oplus N_{\mu_2} \oplus \cdots. (b) If NN and {μn}\{ \mu_n\} are as in (a) and M≅Nν1⊕Nν2⊕⋯M \cong N_{\nu_1} \oplus N_{\nu_2} \oplus \cdots, where νn+1≪νn\nu_{n+1} \ll \nu_n for all nn, then N≅MN \cong M if and only if [μn]=[νn][\mu_n] = [\nu_n] for all nn.

Theorem 10.16: If NN is a normal operator, then there are mutually singular measures μ∞,μ1,μ2,…\mu_\infty , \mu_1, \mu_2, \dots (some of which may be zero) such that

N≅Nμ∞(∞)⊕Nμ1⊕Nμ2(2)⊕⋯ . N \cong N_{\mu _\infty }^{(\infty )} \oplus N_{\mu _1} \oplus N_{\mu _2}^{(2)} \oplus \cdots .

If MM is another normal operator with corresponding measures ν∞,ν1,ν2,…\nu_\infty , \nu_1, \nu_2, \dots, then N≅MN \cong M if and only if [μn]=[νn][\mu_n] = [\nu_n] for 1≤n≤∞1 \leq n \leq \infty.

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Problem IX.10.8

If μ\mu is Lebesgue measure on R\mathbb {R} and N=N = multiplication by eixe^{ix} on L2(μ)L^2(\mu ), show that N≅Nm(∞)N \cong N_m^{(\infty )}, where m=m = arc length measure on ∂D\partial \mathbb {D}.

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Problem IX.10.9

Define U:L2(R)→L2(R)U: L^2(\mathbb {R}) \to L^2(\mathbb {R}) by (Uf)(t)=f(t−1)(Uf)(t) = f(t - 1). Show that UU is unitary and find its scalar-valued spectral measure and multiplicity function.

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Problem IX.10.10

Represent NN as in the following theorem (Theorem 10.1(a): if NN is a normal operator, then there is a sequence (possibly finite) of measures {μn}\{ \mu_n\} on C\mathbb {C} such that μn+1≪μn\mu_{n+1} \ll \mu_n for all nn and N≅Nμ1⊕Nμ2⊕⋯N \cong N_{\mu_1} \oplus N_{\mu_2} \oplus \cdots) and find the corresponding representation for N⊕N=N(2)N \oplus N = N^{(2)}; for N(3)N^{(3)}; for N(∞)N^{(\infty )}. (Are you surprised by the result for N(∞)N^{(\infty )}?)

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Problem IX.10.11

Prove the results and solve the exercises from II.8.

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Problem IX.10.12

Let NN be a normal operator and show that N≅N(2)N \cong N^{(2)} if and only if there is a ∗*-cyclic normal operator MM such that N≅M(∞)N \cong M^{(\infty )}. What does this say about the multiplicity function for NN?

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Problem IX.10.13

Let (X,Ω,μ)(X, \Omega , \mu ) be a measure space such that L2(μ)L^2(\mu ) is separable, let ϕ∈L∞(μ)\phi \in L^\infty (\mu ), and let N=MϕN = M_\phi on L2(μ)L^2(\mu ). Find the decompositions of NN obtained in Theorem 10.1 and Theorem 10.16, stated below.

Theorem 10.1: (a) If NN is a normal operator, then there is a sequence (possibly finite) of measures {μn}\{ \mu_n\} on C\mathbb {C} such that μn+1≪μn\mu_{n+1} \ll \mu_n for all nn and N≅Nμ1⊕Nμ2⊕⋯N \cong N_{\mu_1} \oplus N_{\mu_2} \oplus \cdots. (b) If NN and {μn}\{ \mu_n\} are as in (a) and M≅Nν1⊕Nν2⊕⋯M \cong N_{\nu_1} \oplus N_{\nu_2} \oplus \cdots, where νn+1≪νn\nu_{n+1} \ll \nu_n for all nn, then N≅MN \cong M if and only if [μn]=[νn][\mu_n] = [\nu_n] for all nn.

Theorem 10.16: If NN is a normal operator, then there are mutually singular measures μ∞,μ1,μ2,…\mu_\infty , \mu_1, \mu_2, \dots (some of which may be zero) such that

N≅Nμ∞(∞)⊕Nμ1⊕Nμ2(2)⊕⋯ . N \cong N_{\mu _\infty }^{(\infty )} \oplus N_{\mu _1} \oplus N_{\mu _2}^{(2)} \oplus \cdots .

If MM is another normal operator with corresponding measures ν∞,ν1,ν2,…\nu_\infty , \nu_1, \nu_2, \dots, then N≅MN \cong M if and only if [μn]=[νn][\mu_n] = [\nu_n] for 1≤n≤∞1 \leq n \leq \infty.

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Problem IX.10.14

Let μ\mu be a compactly supported measure on C\mathbb {C}, ϕ\phi a bounded Borel function on C\mathbb {C}, and suppose {Δn}\{ \Delta_n\} are pairwise disjoint Borel sets such that ϕ\phi is one-to-one on each Δn\Delta_n and μ(C∖⋃n=1∞Δn)=0\mu (\mathbb {C} \setminus \bigcup_{n=1}^\infty \Delta_n) = 0. Let ϕn=ϕχΔn\phi_n = \phi \chi_{\Delta_n} and μn=μ∘ϕn−1\mu_n = \mu \circ \phi_n^{-1} for n≥1n \geq 1. Prove that MϕM_\phi on L2(μ)L^2(\mu ) is unitarily equivalent to ⨁n=1∞Nμn\bigoplus_{n=1}^\infty N_{\mu_n}.

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