Linear Maps on Hilbert Spaces

Define $T: \ell^{2} \rightarrow \ell^{2}$ by $T\left(a_{1}, a_{2}, \ldots \right) = \left(0, a_{1}, a_{2}, \ldots \right)$. Find a formula for $T^{*}$.

Suppose $V$ is a Hilbert space, $U$ is a closed subspace of $V$, and $T: U \rightarrow V$ is defined by $T f = f$. Describe the linear operator $T^{*}: V \rightarrow U$.

Suppose $V$ and $W$ are Hilbert spaces and $g \in V, h \in W$. Define $T \in \mathcal{B}(V, W)$ by $T f = \left\langle f, g \right\rangle h$. Find a formula for $T^{*}$.

Suppose $V$ and $W$ are Hilbert spaces and $T \in \mathcal{B}(V, W)$ has finite-dimensional range. Prove that $T^{*}$ also has finite-dimensional range.

Prove or give a counterexample: If $V$ is a Hilbert space and $T: V \rightarrow V$ is a bounded linear map such that $\operatorname {dim}\operatorname {null}(T)< \infty$, then $\operatorname {dim}\operatorname {null}(T^{*}) < \infty$.

Suppose $T$ is a bounded linear map from a Hilbert space $V$ to a Hilbert space $W$. Prove that $\left\| T^{*} T\right\| = \left\| T\right\|^{2}$.

Suppose $V$ is a Hilbert space and $\operatorname {Inv}(V)$ is the set of invertible bounded operators on $V$. Think of $\operatorname {Inv}(V)$ as a metric space with the metric it inherits as a subset of $\mathcal{B}(V)$. Show that $T \mapsto T^{-1}$ is a continuous function from $\operatorname {Inv}(V)$ to $\operatorname {Inv}(V)$.

Suppose $T$ is a bounded operator on a Hilbert space.

Suppose $b_{1}, b_{2}, \ldots$ is a bounded sequence in $\mathbf{F}$. Define a bounded linear map $T: \ell^{2} \rightarrow \ell^{2}$ by

Suppose $h \in \mathcal{L}^{\infty }(\mathbb {R})$ and $M_{h}: L^{2}(\mathbb {R}) \rightarrow L^{2}(\mathbb {R})$ is the bounded operator defined by $M_{h} f = f h$.

Define $T: \ell^{2} \rightarrow \ell^{2}$ by $T\left(a_{1}, a_{2}, a_{3}, \ldots \right) = \left(a_{2}, a_{3}, a_{4}, \ldots \right)$. Suppose $\alpha \in \mathbf{F}$.

Suppose $V$ is a Hilbert space.

Suppose $T$ is a bounded operator on a Hilbert space $V$.

Verify all the assertions in Example 10.33.

Suppose $T$ is a bounded operator on a Hilbert space $V$.

Suppose $E$ is a bounded subset of $\mathbf{F}$. Show that there exists a Hilbert space $V$ and $T \in \mathcal{B}(V)$ such that the set of eigenvalues of $T$ equals $E$.

Suppose $E$ is a nonempty closed bounded subset of $\mathbf{F}$. Show that there exists $T \in \mathcal{B}\left(\ell^{2}\right)$ such that $\operatorname {sp}(T) = E$.

Give an example of a bounded operator $T$ on a normed vector space such that for every $\alpha \in \mathbf{F}$, the operator $T-\alpha I$ is not invertible.

Suppose $T$ is a bounded operator on a complex nonzero Banach space $V$.

Prove that if $T$ is an operator on a Hilbert space $V$ such that $\langle T f, g\rangle = \langle f, T g\rangle$ for all $f, g \in V$, then $T$ is a bounded operator.

Suppose $P$ is a bounded operator on a Hilbert space $V$ such that $P^{2} = P$. Prove that $P$ is self-adjoint if and only if there exists a closed subspace $U$ of $V$ such that $P = P_{U}$.

Suppose $V$ is a real Hilbert space and $T \in \mathcal{B}(V)$. The complexification of $T$ is the function $T_{\mathbb {C}}: V_{\mathbb {C}} \rightarrow V_{\mathbb {C}}$ defined by

for $f, g \in V$ (see Exercise 4 in Section 8B for the definition of $V_{\mathbb {C}}$).

Suppose $T$ is a bounded operator on a Hilbert space $V$ such that $\left\langle Tf, f \right\rangle \geq 0$ for all $f \in V$. Prove that $\operatorname {sp}(T) \subseteq [0, \infty )$.

Suppose $P$ is a bounded operator on a Hilbert space $V$ such that $P^{2} = P$. Prove that $P$ is self-adjoint if and only if $P$ is normal.

Prove that a normal operator on a separable Hilbert space has at most countably many eigenvalues.

Prove or give a counterexample: If $T$ is a normal operator on a Hilbert space and $T = A+i B$, where $A$ and $B$ are self-adjoint, then $\left\| T\right\| = \sqrt{\left\| A\right\|^{2}+\left\| B\right\|^{2}}$.

Suppose $T$ is a normal operator on a Hilbert space and $\alpha \in \mathbb {F}$. Prove that $\alpha \in \operatorname {sp}(T)$ if and only if $\alpha$ is an approximate eigenvalue of $T$.

Suppose $T$ is a normal operator on a Hilbert space.

Suppose $\left\{ e_{k}\right\}_{k \in \mathbb {N}}$is an orthonormal basis of a Hilbert space $V$. Suppose also that $T$ is a normal operator on $V$ and $e_{k}$ is an eigenvector of $T$ for every $k \geq 2$. Prove that $e_{1}$ is an eigenvector of $T$.

Prove that if $T$ is a self-adjoint operator on a Hilbert space, then $\left\| T^{n}\right\| = \left\| T\right\|^{n}$ for every $n \in \mathbb {N}$.

Prove that if $T$ is a normal operator on a Hilbert space, then $\left\| T^{n}\right\| = \left\| T\right\|^{n}$ for every $n \in \mathbb {N}$.

Suppose $T$ is an invertible operator on a Hilbert space. Prove that $T$ is unitary if and only if $\left\| T\right\| = \left\| T^{-1}\right\| = 1$.

Suppose $T$ is a bounded operator on a complex Hilbert space, with $T = A+i B$, where $A$ and $B$ are self-adjoint (see 10.54). Prove that $T$ is unitary if and only if $T$ is normal and $A^{2}+B^{2} = I$.

Suppose $T$ is a unitary operator on a complex Hilbert space such that $T-I$ is invertible. Prove that

is a self-adjoint operator.

Suppose $T$ is a self-adjoint operator on a complex Hilbert space. Prove that

is a unitary operator.

For $T$ a bounded operator on a Banach space, define $e^{T}$ by

Suppose $(X, \mathcal{S}, \mu )$ is a $\sigma$-finite measure space and $h \in L^{\infty }(\mu )$. As usual, let $M_{h} \in \mathcal{B}\left(L^{2}(\mu )\right)$ denote the multiplication operator defined by $M_{h} f = f h$. Prove that $M_{h}$ is a partial isometry if and only if there exists a set $E \in \mathcal{S}$ such that $\left|h\right| = \chi_{E}$.

Suppose $T$ is an isometry on a Hilbert space. Prove that $T^{*}$ is a partial isometry.

Suppose $T$ is a bounded operator on a Hilbert space $V$. Prove that $T$ is a partial isometry if and only if $T^{*} T = P_{U}$ for some closed subspace $U$ of $V$.

Prove that if $T$ is a compact operator on a Hilbert space $V$ and $e_{1}, e_{2}, \ldots$ is an orthonormal sequence in $V$, then $\lim_{n \rightarrow \infty } T e_{n} = 0_V$.

Prove that if $T$ is a compact operator on $L^{2}([0,1])$, then $\lim_{n \rightarrow \infty } \sqrt{n}\left\| T\left(x^{n}\right)\right\|_{2} = 0$, where $x^{n}$ means the element of $L^{2}([0,1])$ defined by $x \mapsto x^{n}$.

Suppose $T$ is a compact operator on a Hilbert space $V$ and $f_{1}, f_{2}, \ldots$ is a sequence in $V$ such that $\lim_{n \rightarrow \infty } \left\langle f_{n}, g \right\rangle = 0$ for every $g \in V$. Prove that $\lim_{n \rightarrow \infty }\left\| T f_{n}\right\| = 0$.

Suppose $h \in L^{\infty }(\mathbb {R})$. Define $M_{h} \in \mathcal{B}\left(L^{2}(\mathbb {R})\right)$ by $M_{h} f = f h$. Prove that if $\left\| h\right\|_{\infty } > 0$, then $M_{h}$ is not compact.

Suppose $\left(b_{1}, b_{2}, \ldots \right) \in \ell^{\infty }$. Define $T: \ell^{2} \rightarrow \ell^{2}$ by

Prove that $T$ is compact if and only if $\lim_{n \rightarrow \infty } b_{n} = 0$.

Suppose $T$ is a bounded operator on a Hilbert space $V$. Prove that if there exists an orthonormal basis $\left\{ e_{k}\right\}_{k \in \Gamma }$ of $V$ such that

then $T$ is compact.

Suppose $T$ is a bounded operator on a Hilbert space $V$. Prove that if $\left\{ e_{k}\right\}_{k \in \Gamma }$ and $\left\{ f_{j}\right\}_{j \in \Omega }$ are orthonormal bases of $V$, then

Suppose $T$ is a bounded operator on a Hilbert space. Prove that $T$ is compact if and only if $T^{*} T$ is compact.

Prove that if $T$ is a compact operator on an infinite-dimensional Hilbert space, then $\left\| I-T\right\| \geq 1$.

Show that if $T$ is a surjective but not injective operator on a vector space $V$, then

Suppose $T$ is a compact operator on a Hilbert space and $\alpha \in \mathbf{F} \backslash \left\{ 0\right\}$.

Prove that if $f:[0,1] \rightarrow \mathbf{F}$ is a continuous function, then there exists a continuous function $g:[0,1] \rightarrow \mathbf{F}$ such that

for all $x \in [0,1]$.

Suppose $S$ is a bounded invertible operator on a Hilbert space $V$ and $T$ is a compact operator on $V$.

Suppose $T$ is a compact operator on a Hilbert space $V$. Prove that range $T$ is a separable subspace of $V$.

Suppose $T$ is a compact operator on a Hilbert space $V$ and $e_{1}, e_{2}, \ldots$ is an orthonormal basis of range $T$. Let $P_{n}$ denote the orthogonal projection of $V$ onto $\operatorname {span}\left\{ e_{1}, \ldots , e_{n}\right\}$.

Prove that if $T$ is a compact operator on a Hilbert space $V$, then there exists a sequence $S_{1}, S_{2}, \ldots$ of invertible operators on $V$ such that $\lim_{n \rightarrow \infty }\left\| T-S_{n}\right\| = 0$.

Suppose $T$ is a bounded operator on a Hilbert space such that $p(T)$ is compact for some nonzero polynomial $p$ with coefficients in $\mathbf{F}$. Prove that $\operatorname {sp}(T)$ is a countable set.

Suppose $T \in \mathcal{B}\left(\ell^{2}\right)$ is defined by $T\left(a_{1}, a_{2}, a_{3}, \ldots \right) = \left(a_{2}, a_{3}, a_{4}, \ldots \right)$. Suppose also that $\alpha \in \mathbf{F}$ and $\left|\alpha \right| < 1$.

Prove that the geometric multiplicity of an eigenvalue of a normal operator on a Hilbert space equals the algebraic multiplicity of that eigenvalue.

Prove that every nonzero eigenvalue of a compact operator on a Hilbert space has finite algebraic multiplicity.

Prove that if $T$ is a compact operator on a Hilbert space and $\alpha$ is a nonzero eigenvalue of $T$, then the algebraic multiplicity of $\alpha$ as an eigenvalue of $T$ equals the algebraic multiplicity of $\overline{\alpha }$ as an eigenvalue of $T^{*}$.

Prove that if $V$ is a separable Hilbert space, then $\mathcal{C}(V)$, the Banach space of compact operators on $V$, is separable.

Prove that if $T$ is a compact operator on a nonzero Hilbert space, then $\left\| T\right\|^{2}$ is an eigenvalue of $T^{*} T$.

Prove that if $T$ is a self-adjoint operator on a nonzero Hilbert space $V$, then

Suppose $T$ is a bounded operator on a Hilbert space $V$ and $U$ is a closed subspace of $V$. Prove that the following are equivalent:

Suppose $T$ is a bounded operator on a Hilbert space $V$ and $U$ is a closed subspace of $V$. Prove that the following are equivalent:

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 $V$.

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 $V$.

Suppose $T$ is a self-adjoint compact operator on a Hilbert space that has only finitely many distinct eigenvalues. Prove that $T$ has finite-dimensional range.

Suppose $T$ is a compact normal operator on a nonzero Hilbert space $V$. Prove that there is a subspace of $V$ with dimension 1 or 2 that is an invariant subspace for $T$.

Suppose $T$ is a self-adjoint compact operator on a Hilbert space and $\left\| T\right\| \leq \frac{1}{4}$. Prove that there exists a self-adjoint compact operator $S$ such that $S^{2}+S = T$.

For $k \in \mathbb {Z}$, define $g_{k} \in L^{2}((-\pi , \pi ])$ and $h_{k} \in L^{2}((-\pi , \pi ])$ by

here we are assuming that $\mathbf{F} = \mathbb {C}$.

Suppose $T$ is a compact operator on a Hilbert space. Prove that $s_{1}(T) = \left\| T\right\|$.

Suppose $T$ is a compact operator on a Hilbert space and $n \in \mathbb {N}$. Prove that $\operatorname {dim}\operatorname {range}T < n$ if and only if $s_{n}(T) = 0$.

Suppose $T$ is a compact operator on a Hilbert space $V$ with singular value decomposition

for all $f \in V$. For $n \in \mathbb {N}$, define $T_{n}: V \rightarrow V$ by

Prove that $\lim_{n \rightarrow \infty }\left\| T-T_{n}\right\| = 0$.

Suppose $T$ is a compact operator on a Hilbert space $V$ and $n \in \mathbb {N}$. Prove that