Unbounded Operators

If $A$, $B$, and $AB$ are densely defined linear operators, show that $(AB)^* \supseteq B^*A^*$.

Let $e_0, e_1, \ldots$ be an orthonormal basis for $\mathcal{H}$ and let $\alpha_0, \alpha_1, \ldots$ be complex numbers. Define

and let $Ah = \sum_{n=0}^{\infty } \alpha_n \left\langle h, e_n \right\rangle e_n$ for $h$ in $\mathcal{D}$. Verify that $A \in \mathcal{C}(\mathcal{H})$ with $\operatorname {dom} A = \mathcal{D}$, that $\operatorname {dom} A^* = \mathcal{D}$, and that $A^*h = \sum_{n=0}^{\infty } \overline{\alpha_n} \left\langle h, e_n \right\rangle e_n$ for all $h$ in $\mathcal{D}$.

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $\phi : X \to \mathbb {C}$ be an $\Omega$-measurable function. Let $\mathcal{D} = \left\{ f \in L^2(\mu ) : \phi f \in L^2(\mu )\right\}$ and define $Af = \phi f$ for all $f$ in $\mathcal{D}$. Verify that $A \in \mathcal{C}(L^2(\mu ))$, that $\operatorname {dom} A^* = \mathcal{D}$, and that $A^*f = \overline{\phi } f$ for $f$ in $\mathcal{D}$.

Define an unbounded weighted shift and determine its adjoint.

Let $\mathcal{D}$ be all functions $f : [0,1] \to \mathbb {C}$ that are absolutely continuous with $f' \in L^2(0,1)$ and such that $f(0) = f(1) = 0$. Define $A : L^2(0,1) \to L^2(0,1)$ by $Af = if'$ for $f$ in $\mathcal{D}$.

Verify that $\mathcal{D}$ is dense in $L^2(0,1)$, that $A \in \mathcal{C}(L^2(0,1))$, and that $\operatorname {dom} A^* = \left\{ g : g \text{ is absolutely continuous on } [0,1], \, g' \in L^2(0,1)\right\}$, with $A^*g = ig'$ for $g$ in $\operatorname {dom} A^*$.

If $\mathcal{H}$ is infinite dimensional, show that there is a linear operator $A : \mathcal{H} \to \mathcal{H}$ such that $\operatorname {gra} A$ is dense in $\mathcal{H} \oplus \mathcal{H}$. What does this say about $\operatorname {dom} A^*$? (See Lindsay [1984].)

Let $\mathcal{D}$ be the set of absolutely continuous functions $f$ such that $f' \in L^2(0,1)$. Let $Df = f'$ for $f$ in $\mathcal{D}$ and let $(Af)(x) = xf(x)$ for $f$ in $L^2(0,1)$. Show that $DA - AD \subseteq 1$.

If $\mathcal{A}$ is a Banach algebra with identity, show that there are no elements $a, b$ in $\mathcal{A}$ such that $ab - ba = 1$.

Let $A \in \mathcal{C}(\mathcal{H})$. Prove:

Define $A : L^2(\mathbb {R}) \to L^2(\mathbb {R})$ by $(Af)(x) = \exp (-x^2) f(x-1)$ for all $f$ in $L^2(\mathbb {R})$.

If $A \in \mathcal{C}(\mathcal{H})$, show that $A^*A \in \mathcal{C}(\mathcal{H})$. Show that $-1 \notin \sigma (A^*A)$ and that if $B = (1 + A^*A)^{-1}$, $\left\| B\right\| \leq 1$.

If $B$ is the bounded operator obtained in Exercise 11, show that $C = AB$ is also bounded and $\left\| C\right\| \leq 1$.

If $A$ is a self-adjoint operator, then $\lambda \in \rho (A)$ if and only if $A - \lambda$ is surjective.

If $A$ is symmetric, show that all of the eigenvalues of $A$ are real.

If $A$ is symmetric and $\lambda , \mu$ are distinct eigenvalues, show that $\ker (A - \lambda ) \perp \ker (A - \mu )$.

Show that the closure of a symmetric operator is symmetric.

Let $\mathcal{D} = \left\{ f \in L^2(0, \infty ) : \text{for every } c > 0, \, f \text{ is absolutely continuous on } [0, c], \, f(0) = 0, \text{ and } f' \in L^2(0, \infty )\right\}$. Define $Af = if'$ for $f$ in $\mathcal{D}$. Show that $A$ is a densely defined closed operator and find $\operatorname {dom} A^*$. Show that $A$ is symmetric with deficiency indices $n_+ = 0$ and $n_- = 1$.

Let $\mathcal{E} = \left\{ f \in L^2(-\infty , 0) : \text{for every } c < 0, \, f \text{ is absolutely continuous on } [c, 0], \, f(0) = 0, \text{ and } f' \in L^2(-\infty , 0)\right\}$. Define $Af = if'$ for $f$ in $\mathcal{E}$. Show that $A$ is a densely defined closed operator and find $\operatorname {dom} A^*$. Show that $A$ is symmetric with deficiency indices $n_+ = 1$, $n_- = 0$.

If $k, l$ are any nonnegative integers or $\infty$, show that there is a closed symmetric operator $A$ with $n_+ = k$ and $n_- = l$.

Let $C_c^{(2)}(0,1)$ be all twice continuously differentiable functions on $(0,1)$ with compact support and let $Af = -f''$ for $f$ in $C_c^{(2)}(0,1)$. Show that the closure of $A$ is a densely defined symmetric operator and determine all of its self-adjoint extensions.

If $A \in \mathcal{C}(\mathcal{H})$, show that $A^*A$ is self-adjoint (see Exercise 1.11).

Say that an operator $A$ is positive if $\left\langle Ah, h \right\rangle \geq 0$ for all $h$ in $\operatorname {dom} A$. Prove that if $A$ is positive and self-adjoint, then $\sigma (A) \subseteq [0, \infty )$. If $A$ is only assumed to be closed and positive, show that this conclusion may fail.

(Lasser [1972]) Let $\mathcal{M}$ be a dense linear manifold in $\mathcal{H}$ and let $\mathcal{A}$ consist of all linear transformations $A$ such that $\operatorname {dom} A = \mathcal{M}$, $A\mathcal{M} \subseteq \mathcal{M}$, the adjoint of $A$ exists, $\mathcal{M} \subseteq \operatorname {dom} A^*$, and $A^*\mathcal{M} \subseteq \mathcal{M}$. Prove that $A \mapsto A^*\rvert \mathcal{M}$ defines an involution on $\mathcal{A}$.

If $U$ is a partial isometry, show that the following statements are equivalent:

1. $\ker (1 - U) = (0)$;

2. $\ker (1 - U^*) = (0)$;

3. $\operatorname {ran}(1 - U)$ is dense;

4. $\operatorname {ran}(1 - U^*)$ is dense.

Let $U$ be a partial isometry with initial and final spaces $\mathcal{M}$ and $\mathcal{N}$, respectively. Show that the following statements are equivalent:

1. $(1 - U)\mathcal{M}$ is dense;

2. $(1 - U^*)\mathcal{N}$ is dense;

3. $\ker (U^* - U^*U) = (0)$;

4. $\ker (U - UU^*) = (0)$.

Find a partial isometry $U$ such that $\ker (1 - U) = (0)$ but $(1 - U)(\ker U)^{\perp }$ is not dense.

If $A$ is a densely defined closed symmetric operator and $B$ and $C$ are the operators defined in Exercises 1.11 and 1.12, then the Cayley transform of $A$ is an extension of $(C - iB)(C + iB)^{-1}$.

Let $e_0, e_1, \ldots$ be an orthonormal basis for $\mathcal{H}$ and let $\alpha_0, \alpha_1, \ldots$ be complex numbers. Define

and let $Ah = \sum_{n=0}^{\infty } \alpha_n \left\langle h, e_n \right\rangle e_n$ for $h$ in $\mathcal{D}$. Find the Cayley transform of $A$ when each $\alpha_n$ is real.

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $\phi : X \to \mathbb {C}$ be an $\Omega$-measurable function. Let $\mathcal{D} = \left\{ f \in L^2(\mu ) : \phi f \in L^2(\mu )\right\}$ and define $Af = \phi f$ for all $f$ in $\mathcal{D}$. Find the Cayley transform of $A$ when $\phi$ is real valued.

Let $S$ be the unilateral shift of multiplicity 1 (see Exercise IX.6.4) and find the symmetric operator $A$ such that $S$ is the Cayley transform of $A$.

Let $U = S^*$, where $S$ is the unilateral shift of multiplicity 1. Is $U$ the Cayley transform of a symmetric operator $A$? If so, find it.

Let $(X, \Omega )$ be a measurable space, $\mathcal{H}$ a Hilbert space, and $E$ a spectral measure for $(X, \Omega , \mathcal{H})$. Let $\Phi (X, \Omega )$ be the algebra of all $\Omega$-measurable functions $\phi : X \to \mathbb {C}$ and define $\rho : \Phi (X, \Omega ) \to \mathcal{C}(\mathcal{H})$ by $\rho (\phi ) = \int \phi \, dE$. Prove that for $\phi , \psi$ in $\Phi (X, \Omega )$:

Show that if $A$ is a symmetric operator that is normal, then $A$ is self-adjoint.

Let $E$ be a spectral measure for $(X, \Omega , \mathcal{H})$, let $\phi : X \to \mathbb {C}$ be an $\Omega$-measurable function, and let

where $E_{h,k}(\Delta ) = \left\langle E(\Delta )h, k \right\rangle$. (Theorem 4.7 shows that for $h$ in $\mathcal{D}_\phi$ and $f$ in $\mathcal{H}$, $\left\langle \left(\int \phi \, dE\right) h, f \right\rangle = \int \phi \, dE_{h,f}$.)

Show that for $h$ in $\mathcal{D}_\phi$, $\left\| \left(\int \phi \, dE\right) h\right\|^2 = \int \left|\phi \right|^2 \, dE_{h,h}$.

Using the notation of Theorem 4.10 (see X.4.1 above: $\rho (\phi ) = \int \phi \, dE$ for $E$ a spectral measure and $\phi$ an $\Omega$-measurable function), what is $\sigma \left(\int \phi \, dE\right)$?

Recall the construction from the proof of the Spectral Theorem (4.11): given a normal operator $N$, one lets $B = (1 + N^*N)^{-1} = \int_0^1 t \, dP(t)$ be the spectral decomposition of $B$, puts $P_n = P\left(\left[1/(n+1), 1/n\right]\right)$ and $\mathcal{H}_n = P_n \mathcal{H}$ for $n \geq 1$, so that $\mathcal{H}_n$ reduces $N$ and $N_n \equiv N \rvert \mathcal{H}_n$ is a bounded normal operator with $\sigma (N_n) \subseteq \Delta_n \equiv \left\{ z \in \mathbb {C} : (n-1)^{1/2} \leq \left|z\right| \leq n^{1/2}\right\}$; letting $N_n = \int z \, dE_n(z)$ be the spectral decomposition of $N_n$, the spectral measure of $N$ is then defined by $E(\Delta ) = \sum_{n=1}^{\infty } E_n(\Delta \cap \Delta_n)$.

If $\Delta_n$ and $E_n$ are as in this construction, show that $E_n(\Delta_{n+1}) = E_n(\Delta_{n-1}) = 0$.

Use the Spectral Theorem to show that if $0 < a \leq b < \infty$, $\Delta = \left\{ z \in \mathbb {C} : a \leq \left|z\right| \leq b\right\}$, and $N = \int z \, dE(z)$ is the spectral decomposition of the normal operator $N$, then

State and prove a polar decomposition for operators in $\mathcal{C}(\mathcal{H}, \mathcal{K})$.

If $A$ is self-adjoint, prove that $\exp (iA)$ is unitary.

If $N$, $M$ are normal operators and $A$ is a bounded operator such that $AN \subseteq MA$, then $AN^* \subseteq M^*A$.

Recall that if $N$ is a normal operator on $\mathcal{H}$, a vector $e_0$ is a star-cyclic vector for $N$ if for all non-negative integers $k$ and $l$, $e_0 \in \operatorname {dom}(N^{*k}N^l)$ and $\mathcal{H} = \bigvee \left\{ N^{*k}N^l e_0 : k, l \geq 0\right\}$.

Prove: if $N$ is a normal operator on $\mathcal{H}$ with a star-cyclic vector $e_0$, then there is a finite measure $\mu$ on $\mathbb {C}$ such that every polynomial in $z$ and $\overline{z}$ belongs to $L^2(\mu )$ and there is an isomorphism $W : \mathcal{H} \to L^2(\mu )$ such that $We_0 = 1$ and $WNW^{-1} = N_\mu$.

Let $\mu$ be a finite measure on $\mathbb {C}$ such that every polynomial in $z$ and $\overline{z}$ belongs to $L^2(\mu )$ and the collection of such polynomials is dense in $L^2(\mu )$. Let $\mathcal{D}_\mu = \left\{ f \in L^2(\mu ) : zf \in L^2(\mu )\right\}$ and define $N_\mu f = zf$ for $f$ in $\mathcal{D}_\mu$. (Recall that $N_\mu$ is a normal operator and $1$ is a star-cyclic vector for $N_\mu$.)

If $\mu_1, \mu_2$ are finite measures on $\mathbb {C}$ satisfying the conditions above and $N_{\mu_1}, N_{\mu_2}$ are defined accordingly, show that $N_{\mu_1} \cong N_{\mu_2}$ iff $[\mu_1] = [\mu_2]$.

Recall Theorem 4.19: if $N$ is a normal operator on the separable Hilbert space $\mathcal{H}$, then there is a $\sigma$-finite measure space $(X, \Omega , \mu )$ and an $\Omega$-measurable function $\phi$ such that $N$ is unitarily equivalent to $M_\phi$ on $L^2(\mu )$.

The proof is only sketched in the text as follows. Write $N$ as the (unbounded) direct sum of bounded normal operators $\left\{ N_n\right\}$. For each $n$ there is a $\sigma$-finite measure space $(X_n, \Omega_n, \mu_n)$ and a bounded $\Omega_n$-measurable function $\phi_n$ such that $N_n \cong M_{\phi_n}$. Let $X$ be the disjoint union of $\left\{ X_n\right\}$ and let $\Omega = \left\{ \Delta \subseteq X : \Delta \cap X_n \in \Omega_n \text{ for every } n\right\}$. If $\Delta \in \Omega$, let $\mu (\Delta ) = \sum_{n=1}^{\infty } \mu_n(\Delta \cap X_n)$. Let $\phi : X \to \mathbb {C}$ be defined by $\phi (x) = \phi_n(x)$ if $x \in X_n$. Then $\phi$ is $\Omega$-measurable and $N \cong M_\phi$ on $L^2(X, \Omega , \mu )$.

Fill in the details of this proof.

If $U : \mathbb {R} \to \mathcal{B}(\mathcal{H})$ is such that $U(t)$ is unitary for all $t$, $U(s+t) = U(s)U(t)$ for all $s, t$, and $U : \mathbb {R} \to (\mathcal{B}(\mathcal{H}), \mathrm{WOT})$ is continuous, then $U$ is SOT-continuous.

Show that for every integer $n$ there is a continuously differentiable function $\phi_n$ such that both $\phi_n$ and $\phi_n' \in L^1(0, \infty )$, $\phi_n(t) = 0$ if $t \geq 1/n$, and $\int_0^{\infty } \phi_n(t) \, dt = 1$.

Recall the construction from the proof of Stone's Theorem. Let $U$ be a strongly continuous one-parameter unitary group on $\mathcal{H}$, and let $\mathcal{D}$ be the set of all vectors $h$ in $\mathcal{H}$ such that $\lim_{t \to 0} t^{-1}[U(t)h - h]$ exists ($\mathcal{D}$ is shown to be dense in $\mathcal{H}$ using the operators $T_\phi h = \int_0^{\infty } \phi (t) U(t) h \, dt$ for continuous $\phi \in L^1(0, \infty )$, together with functions $\phi_n$ as in Exercise 2). For $h$ in $\mathcal{D}$, define

Prove that $A$ is symmetric.

Adopt the following notation from the proof of Stone's Theorem. Let $U$ be a strongly continuous one-parameter unitary group on $\mathcal{H}$ and let $\mathcal{L}$ be the collection of all continuous functions $\phi$ on $\mathbb {R}$ such that $\phi \in L^1(0, \infty )$. For $\phi$ in $\mathcal{L}$ and $h$ in $\mathcal{H}$, define

and let $A$ be the infinitesimal generator of $U$. Let $\phi , \psi \in \mathcal{L}$ and show:

Let $U$ be a strongly continuous one-parameter unitary group with infinitesimal generator $A$. Suppose $e$ is a nonzero vector in $\mathcal{H}$ such that $Ae = \lambda e$. What is $U(t)e$? Conversely, suppose there is a nonzero $t$ such that $U(t)$ has an eigenvector. What can be said about $A$? $U(s)$?

(This exercise is designed to give another proof of Proposition 5.15 as well as give additional information.)

Let $U$ be a strongly continuous one-parameter unitary group with infinitesimal generator $A$. Show that if $\left\| U(t) - 1\right\| \to 0$ as $t \to 0$, then as $t \to 0$, $t^{-1} \int_a^{a+t} U(s) \, ds \to U(a)$ in norm. From here show that as $t \to 0$, $t^{-1}[U(t) - 1]$ has a norm limit, and hence $A$ is a bounded operator since it is the norm limit of bounded operators.

Let $\mathcal{D} = \left\{ f \in L^2(\mathbb {R}) : f \text{ is absolutely continuous on every bounded interval in } \mathbb {R} \text{ and } f' \in L^2(\mathbb {R})\right\}$. (Recall that for $f$ in $\mathcal{D}$, letting $Af = if'$ makes $A$ self-adjoint.)

Show that for every $f$ in $\mathcal{D}$ there is a sequence $\left\{ f_n\right\}$ in $C_c^{(1)}(\mathbb {R})$ such that $f_n \to f$ and $f_n' \to f'$ in $L^2(\mathbb {R})$.

Show that the Schwartz space $\mathcal{S}$ with the seminorms $\left\{ \left\| \cdot \right\|_{m,n} : m, n \geq 0\right\}$ is a Fréchet space.

If $\phi$ is infinitely differentiable on $\mathbb {R}$, show that $\phi \in \mathcal{S}$ if and only if for every integer $n \geq 0$ and every polynomial $p$, $\phi^{(n)}(x)p(x) \to 0$ as $\left|x\right| \to \infty$.

If $f \in L^p(\mathbb {R})$, $1 \leq p \leq \infty$, and $g \in L^1(\mathbb {R})$, show that $f * g \in L^p(\mathbb {R})$ and $\left\| f * g\right\|_p \leq \left\| f\right\|_p \left\| g\right\|_1$.

Recall Proposition 6.13: if $\psi \in L^1(\mathbb {R})$ is such that $(2\pi )^{-1/2}\int_{\mathbb {R}} \psi (x) \, dx = 1$ and if for $\varepsilon > 0$, $\psi_\varepsilon (x) = \varepsilon^{-1}\psi (x/\varepsilon )$, then for every $f$ in $C_0(\mathbb {R})$, $\psi_\varepsilon * f(x) \to f(x)$ uniformly on $\mathbb {R}$.

Let $\psi$ and $\psi_\varepsilon$ be as above. If $f \in L^p(\mathbb {R})$, $1 \leq p < \infty$, show that $\left\| f * \psi_\varepsilon - f\right\|_p \to 0$ as $\varepsilon \to 0$. If $f \in L^{\infty }(\mathbb {R})$, show that $f * \psi_\varepsilon \to f$ (weak*).

If $f \in L^1(\mathbb {R})$ and $\hat{f} \in L^1(\mathbb {R})$, show that $f(x) = (2\pi )^{-1/2} \int_{\mathbb {R}} \hat{f}(t) e^{ixt} \, dt$ a.e.

If $\mathscr {F} : L^2(\mathbb {R}) \to L^2(\mathbb {R})$ is the Plancherel Transform and $f \in L^2(\mathbb {R})$, show that $(\mathscr {F}^{-1}f)(x) = (\mathscr {F}f)(-x)$.

Show that $\mathscr {F}^4 = 1$ but $\mathscr {F}^2 \neq 1$. What does this say about $\sigma (\mathscr {F})$?

Find the Fourier transform of the Hermite polynomials. What do you think?

Let $\left\{ m_n : n \geq 0\right\}$ be a sequence of real numbers and show that the following statements are equivalent.

1. There is a positive regular Borel measure $\mu$ on $[0, \infty )$ such that $m_n = \int t^n \, d\mu (t)$ for all $n \geq 0$.

2. If $\alpha_0, \ldots , \alpha_n \in \mathbb {C}$, then $\sum_{j,k=0}^{n} m_{j+k} \alpha_j \overline{\alpha_k} \geq 0$ and $\sum_{j,k=0}^{n} m_{j+k+1} \alpha_j \overline{\alpha_k} \geq 0$.

3. There is a self-adjoint operator $A$ with $\sigma (A) \subseteq [0, \infty )$ and a vector $e$ in $\operatorname {dom} A^n$ for all $n \geq 0$ such that $m_n = \left\langle A^n e, e \right\rangle$ for $n \geq 0$.

Let $m : \mathbb {R} \to \mathbb {C}$ be a function and show that the following statements are equivalent.

1. There is a finite positive measure $\mu$ on $\mathbb {R}$ such that $m(t) = \int e^{ixt} \, d\mu (x)$ for all $t$ in $\mathbb {R}$.

2. $m$ is continuous and if $\alpha_0, \ldots , \alpha_n \in \mathbb {C}$ and $t_0, \ldots , t_n \in \mathbb {R}$, then $\sum_{j,k=0}^{n} m(t_j - t_k) \alpha_j \overline{\alpha_k} \geq 0$.

3. There is a strongly continuous one-parameter unitary group $U(t)$ and a vector $e$ such that $m(t) = \left\langle U(t)e, e \right\rangle$ for all $t$.

Let $\left\{ m_n : n \in \mathbb {Z}\right\} \subseteq \mathbb {C}$ and show that the following statements are equivalent.

1. There is a positive measure $\mu$ on $\partial \mathbb {D}$ such that $m_n = \int z^n \, d\mu (z)$ for all $n$ in $\mathbb {Z}$.

2. If $\alpha_{-n}, \ldots , \alpha_{-1}, \alpha_0, \alpha_1, \ldots , \alpha_n \in \mathbb {C}$, then $\sum_{j,k=-n}^{n} m_{j-k} \alpha_j \overline{\alpha_k} \geq 0$.

3. There is a unitary operator $U$ and a vector $e$ such that $m_n = \left\langle U^n e, e \right\rangle$ for all $n$.

Recall the proof that (7.1b) implies (7.1c) in the Stieltjes moment problem (Theorem 7.1): given a sequence $\left\{ m_n : n \geq 0\right\}$ of real numbers satisfying $\sum_{j,k=0}^{n} m_{j+k}\alpha_j\overline{\alpha_k} \geq 0$ and $\sum_{j,k=0}^{n} m_{j+k+1}\alpha_j\overline{\alpha_k} \geq 0$ for all $\alpha_0, \ldots , \alpha_n \in \mathbb {C}$, one lets $\mathcal{H}_0$ be the vector space of all finitely nonzero sequences $\left\{ \alpha_n : n \geq 0\right\}$ of complex numbers, with the semi-inner product $[x, y] = \sum_{j,k=0}^{\infty } m_{j+k} \alpha_j \overline{\beta_k}$ for $x = \left\{ \alpha_n\right\}$, $y = \left\{ \beta_n\right\}$ in $\mathcal{H}_0$; lets $T_0 x = \left\{ 0, \alpha_0, \alpha_1, \ldots \right\}$ for $x = \left\{ \alpha_n\right\}$ in $\mathcal{H}_0$, so that $[T_0 x, y] = [x, T_0 y]$; lets $\mathcal{K}_0 = \left\{ x \in \mathcal{H}_0 : [x,x] = 0\right\}$, shows $T_0 \mathcal{K}_0 \subseteq \mathcal{K}_0$, and lets $T$ be the densely defined symmetric operator induced by $T_0$ on the Hilbert space $\mathcal{H}$ obtained by completing $\mathcal{H}_1 = \mathcal{H}_0/\mathcal{K}_0$ with respect to the inner product induced by $[\cdot ,\cdot ]$; uses the conjugation $J_0(\left\{ \alpha_n\right\} ) = \left\{ \overline{\alpha_n}\right\}$ on $\mathcal{H}_0$ (which satisfies $J_0 \mathcal{K}_0 \subseteq \mathcal{K}_0$, so induces a conjugation $J$ on $\mathcal{H}$ satisfying the hypotheses of Proposition 7.2) to produce a self-adjoint extension $A$ of $T$; and takes $e_0 = \left\{ 1, 0, 0, \ldots \right\} \in \mathcal{H}_0$ and $e = e_0 + \mathcal{K}_0 \in \mathcal{H}_1$, so that $e \in \operatorname {dom} T^n \subseteq \operatorname {dom} A^n$ for all $n \geq 0$ and $\left\langle A^n e, e \right\rangle = [T_0^n e_0, e_0] = m_n$ for all $n \geq 0$.