Invariant subspace lattices of operators on Banach spaces

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Problem 10D.5 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 10D "Spectral Theorem for Compact Operators"

Suppose $T$ is a bounded operator on a nonseparable normed vector space $V$ . Prove that $T$ has a closed invariant subspace other than $\left\{ 0\right\}$ and $V$ .

Problem 10D.6 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 10D "Spectral Theorem for Compact Operators"

Suppose $T$ is an operator on a Banach space $V$ with dimension greater than 2. Prove that $T$ has an invariant subspace other than $\left\{ 0\right\}$ and $V$ .

Remark.

For this exercise, $T$ is not assumed to be bounded and the invariant subspace is not required to be closed.

Problem VI.4.1 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VI.4 "Invariant Subspaces"

Let $A, B, T \in \mathcal{B}(\mathscr {X})$ such that $TA = BT$ . Show that $\mathrm{graph}(T) \in \mathrm{Lat}(A \oplus B)$ .

Problem VI.4.2 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VI.4 "Invariant Subspaces"

Prove that $\mathscr {M} \in \mathrm{Lat}\ T$ if and only if $\mathscr {M}^\perp \in \mathrm{Lat}\ T^*$ . What does the map $\mathscr {M} \mapsto \mathscr {M}^\perp$ of $\mathrm{Lat}\ T$ into $\mathrm{Lat}\ T^*$ do to the lattice operations?

Problem VI.4.7 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VI.4 "Invariant Subspaces"

Show that if $T \in \mathcal{B}(\mathscr {X})$ and $\mathscr {X}$ is not separable, then $T$ has a nontrivial invariant subspace.

Problem VI.4.8 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VI.4 "Invariant Subspaces"

Give an example of an invertible operator $T$ on a Banach space $\mathscr {X}$ and an invariant subspace $\mathscr {M}$ for $T$ such that $\mathscr {M}$ is not invariant for $T^{-1}$ .

Problem VI.4.9 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VI.4 "Invariant Subspaces"

Let $\mathscr {X}$ be a Banach space over $\mathbb {C}$ , let $K \in \mathcal{B}_0(\mathscr {X})$ and show that if $\mathscr {C}$ is a maximal chain in $\mathrm{Lat}\ K$ , then $\mathscr {C}$ is a maximal chain in the lattice of all subspaces of $\mathscr {X}$ .

Problem VII.4.5 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VII.4 "The Riesz Functional Calculus"

If $\mathcal{X}$ is a Banach space, $A \in \mathcal{B}(\mathcal{X})$ , and $\mathcal{M} \le \mathcal{X}$ such that $(A - \alpha )^{-1} \mathcal{M} \subseteq \mathcal{M}$ for all $\alpha$ in $\rho (A)$ , show that $f(A) \mathcal{M} \subseteq \mathcal{M}$ whenever $f \in \text{Hol}(A)$ .

Problem 156 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 17 "Unilateral Shift"

If $\mathbf{M}_k(\lambda )$ is the orthogonal complement of $\left\{ f_{\lambda }, \ldots , U^{k-1}f_{\lambda }\right\}$ ( $k = 1, 2, 3, \ldots$ ), then $\mathbf{M}_k(\lambda )$ is invariant under $U$ , $\dim \mathbf{M}_k^{\perp }(\lambda ) = k$ , and $\bigvee_{k=1}^{\infty } \mathbf{M}_k^{\perp }(\lambda ) = \mathbf{H}^2$ .

Remark.

Let $U$ be the unilateral shift, with orthonormal basis $\left\{ e_n\right\}$ , $n = 0, 1, 2, \ldots$ , for the underlying space $\mathbf{H}^2$ so that $Ue_n = e_{n+1}$ . Each complex number $\lambda$ of modulus less than $1$ is a simple eigenvalue of $U^{*}$ , with corresponding eigenvector $f_{\lambda } = \sum_{n=0}^{\infty } \lambda^n e_n$ .

Problem 191 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 20 "Examples of Compactness"

If $A$ is the adjoint of a monotone $l^2$ shift, and if $\mathbf{M}$ is a non-trivial subspace invariant under $A$ , then there exists an integer $n$ $(=1,2,3,\ldots )$ such that $\mathbf{M} = \mathbf{M}_n$ .

Remark.

A subspace is invariant under an operator if the operator maps it into itself; "non-trivial" means different from both $0$ and the whole space.

Suppose that $\left\{ \alpha_n\right\}$ is a monotone sequence ( $\alpha_n \geq \alpha_{n+1}$ , $n = 0, 1, 2, \ldots$ ) of positive numbers ( $\alpha_n > 0$ ) such that $\sum_{n=0}^{\infty } \alpha_n^2 < \infty$ . The unilateral weighted shift with the weight sequence $\left\{ \alpha_n\right\}$ is called a monotone $l^2$ shift. The span of the basis vectors $e_n, e_{n+1}, e_{n+2}, \ldots$ is invariant under such a shift, $n = 0, 1, 2, \ldots$ . The orthogonal complement, i.e., the span $\mathbf{M}_n$ of $e_0, \ldots , e_{n-1}$ , is invariant under the adjoint, $n = 1, 2, 3, \ldots$ ; the principal result is that every invariant subspace of that adjoint is one of these orthogonal complements.

Problem 196 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 21 "Subnormal Operators"

Every quasinormal operator on a space of dimension greater than $1$ has a non-trivial invariant subspace.

Remark.

An operator $A$ is called quasinormal if $A$ commutes with $A^{*}A$ . A subspace is invariant under an operator if the operator maps it into itself; "non-trivial" means different from both $0$ and the whole space.

Problem 5.4.7 from Saxe K. "Beginning Functional Analysis" [Sax02]More problems in Section 5.4 "An Introduction to the Invariant Subspace Problem"

(a)(b)

Let $T$ be the Volterra operator on $L^{2}([0,1])$ defined by

T f(s)=\int _{0}^{s} f(t) d t

For $0 \leq \alpha \leq 1$ set

M_{\alpha }=\left\{ f \in L^{2}([0,1]) \mid f(t)=0,0 \leq t \leq \alpha \right\} .

(a)

Prove that each $M_{\alpha }$ is an invariant subspace for $T$ .

(b)

Describe the orthogonal complement of $M_{\alpha }$ in $L^{2}([0,1])$ .

References

[Axl19] Axler S. Measure, Integration & Real Analysis. 3rd ed., Springer, 2019.

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.

[Hal82] Halmos P.R. A Hilbert Space Problem Book. 2nd ed., Springer-Verlag, 1982.

[Sax02] Saxe K. Beginning Functional Analysis. Springer, 2002.