Operators on Inner Product Spaces

Suppose $n$ is a positive integer. Define $T \in \mathcal{L}(\mathbf{F}^{n})$ by

Find a formula for $T^{*}(z_{1}, \ldots , z_{n})$.

Suppose $T \in \mathcal{L}(V, W)$. Prove that

Suppose $T \in \mathcal{L}(V)$ and $\lambda \in \mathbf{F}$. Prove that

Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V$. Prove that

Suppose $T \in \mathcal{L}(V, W)$. Suppose $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$ and $f_{1}, \ldots , f_{m}$ is an orthonormal basis of $W$. Prove that

Suppose $T \in \mathcal{L}(V, W)$. Prove that

Prove that if $T \in \mathcal{L}(V, W)$, then

Suppose $A$ is an $m$-by-$n$ matrix with entries in $\mathbf{F}$. Use (b) in Exercise 7A.7 to prove that the row rank of $A$ equals the column rank of $A$.

Prove that the product of two self-adjoint operators on $V$ is self-adjoint if and only if the two operators commute.

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$. Prove that $T$ is self-adjoint if and only if

for all $v \in V$.

Define an operator $S: \mathbf{F}^{2} \rightarrow \mathbf{F}^{2}$ by $S(w, z) = (-z, w)$.

An operator $B \in \mathcal{L}(V)$ is called skew if

Suppose that $T \in \mathcal{L}(V)$. Prove that $T$ is normal if and only if there exist commuting operators $A$ and $B$ such that $A$ is self-adjoint, $B$ is a skew operator, and $T = A+B$.

Suppose $\mathbf{F} = \mathbb {R}$. Define $\mathcal{A} \in \mathcal{L}(\mathcal{L}(V))$ by $\mathcal{A} T = T^{*}$ for all $T \in \mathcal{L}(V)$.

Define an inner product on $\mathcal{P}_{2}(\mathbb {R})$ by $\left\langle p, q \right\rangle = \int_{0}^{1} p q$. Define an operator $T \in \mathcal{L}(\mathcal{P}_{2}(\mathbb {R}))$ by

Suppose $T \in \mathcal{L}(V)$ is invertible. Prove that

Suppose $\mathbf{F} = \mathbb {R}$.

Suppose $\mathbf{F} = \mathbb {C}$. Show that the set of self-adjoint operators on $V$ is not a subspace of $\mathcal{L}(V)$.

Suppose $\operatorname {dim} V \geq 2$. Show that the set of normal operators on $V$ is not a subspace of $\mathcal{L}(V)$.

Suppose $T \in \mathcal{L}(V)$ and $\left\| T^{*} v\right\| \leq \left\| T v\right\|$ for every $v \in V$. Prove that $T$ is normal.

Suppose $P \in \mathcal{L}(V)$ is such that $P^{2} = P$. Prove that the following are equivalent.

1. $P$ is self-adjoint.

2. $P$ is normal.

3. There is a subspace $U$ of $V$ such that $P = P_{U}$.

Suppose $D: \mathcal{P}_{8}(\mathbb {R}) \rightarrow \mathcal{P}_{8}(\mathbb {R})$ is the differentiation operator defined by $D p = p^{\prime }$. Prove that there does not exist an inner product on $\mathcal{P}_{8}(\mathbb {R})$ that makes $D$ a normal operator.

Give an example of an operator $T \in \mathcal{L}(\mathbb {R}^{3})$ such that $T$ is normal but not self-adjoint.

Suppose $T$ is a normal operator on $V$. Suppose also that $v, w \in V$ satisfy the equations

Show that $\left\| T(v+w)\right\| = 10$.

Suppose $T \in \mathcal{L}(V)$ and

is the minimal polynomial of $T$. Prove that the minimal polynomial of $T^{*}$ is

Suppose $T \in \mathcal{L}(V)$. Prove that $T$ is diagonalizable if and only if $T^{*}$ is diagonalizable.

Fix $u, x \in V$. Define $T \in \mathcal{L}(V)$ by $T v = \left\langle v, u \right\rangle x$ for every $v \in V$.

Suppose $T \in \mathcal{L}(V)$ is normal. Prove that

for every positive integer $k$.

Suppose $T \in \mathcal{L}(V)$ is normal. Prove that if $\lambda \in \mathbf{F}$, then the minimal polynomial of $T$ is not a polynomial multiple of $(x-\lambda )^{2}$.

Prove or give a counterexample: If $T \in \mathcal{L}(V)$ and there is an orthonormal basis $e_{1}, \ldots , e_{n}$ of $V$ such that $\left\| T e_{k}\right\| = \left\| T^{*} e_{k}\right\|$ for each $k = 1, \ldots , n$, then $T$ is normal.

Suppose that $T \in \mathcal{L}(\mathbf{F}^{3})$ is normal and $T(1,1,1) = (2,2,2)$. Suppose $(z_{1}, z_{2}, z_{3}) \in \operatorname {null} T$. Prove that $z_{1}+z_{2}+z_{3} = 0$.

Fix a positive integer $n$. In the inner product space of continuous real-valued functions on $[-\pi , \pi ]$ with inner product $\left\langle f, g \right\rangle = \int_{-\pi }^{\pi } f g$, let

Suppose $T: V \rightarrow W$ is a linear map. Show that under the standard identification of $V$ with $V^{\prime }$ (see 6.58) and the corresponding identification of $W$ with $W^{\prime }$, the adjoint map $T^{*}: W \rightarrow V$ corresponds to the dual map $T^{\prime }: W^{\prime } \rightarrow V^{\prime }$. More precisely, show that

for all $w \in W$, where $\varphi_{w}$ and $\varphi_{T^{*} w}$ are defined as in 6.58.

6.58 (Riesz representation theorem, revisited): suppose $V$ is finite-dimensional. For each $v \in V$, define $\varphi_{v} \in V^{\prime }$ by $\varphi_{v}(u) = \left\langle u, v \right\rangle$ for each $u \in V$. Then $v \mapsto \varphi_{v}$ is a one-to-one function from $V$ onto $V^{\prime }$.

Prove that a normal operator on a complex inner product space is self-adjoint if and only if all its eigenvalues are real.

Suppose $\mathbf{F} = \mathbb {C}$. Suppose $T \in \mathcal{L}(V)$ is normal and has only one eigenvalue. Prove that $T$ is a scalar multiple of the identity operator.

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$ is normal. Prove that the set of eigenvalues of $T$ is contained in $\left\{ 0,1\right\}$ if and only if there is a subspace $U$ of $V$ such that $T = P_{U}$.

Prove that a normal operator on a complex inner product space is skew (meaning it equals the negative of its adjoint) if and only if all its eigenvalues are purely imaginary (meaning that they have real part equal to $0$).

Prove or give a counterexample: If $T \in \mathcal{L}(\mathbb {C}^{3})$ is a diagonalizable operator, then $T$ is normal (with respect to the usual inner product).

Suppose $V$ is a complex inner product space and $T \in \mathcal{L}(V)$ is a normal operator such that $T^{9} = T^{8}$. Prove that $T$ is self-adjoint and $T^{2} = T$.

Give an example of an operator $T$ on a complex vector space such that $T^{9} = T^{8}$ but $T^{2} \neq T$.

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$. Prove that $T$ is normal if and only if every eigenvector of $T$ is also an eigenvector of $T^{*}$.

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$. Prove that $T$ is normal if and only if there exists a polynomial $p \in \mathcal{P}(\mathbb {C})$ such that $T^{*} = p(T)$.

Suppose $V$ is a complex inner product space. Prove that every normal operator on $V$ has a square root.

Prove that every self-adjoint operator on $V$ has a cube root.

Suppose $V$ is a complex vector space and $T \in \mathcal{L}(V)$ is normal. Prove that if $S$ is an operator on $V$ that commutes with $T$, then $S$ commutes with $T^{*}$.

Without using the complex spectral theorem, use the version of Schur's theorem that applies to two commuting operators (take $\mathcal{E} = \left\{ T, T^{*}\right\}$ in Exercise 20 in Section 6B) to give a different proof that if $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$ is normal, then $T$ has a diagonal matrix with respect to some orthonormal basis of $V$.

Exercise 20 in Section 6B (Exercise 6B.20): if $\mathbf{F} = \mathbb {C}$, $V$ is finite-dimensional, and $\mathcal{E} \subseteq \mathcal{L}(V)$ is such that $S T = T S$ for all $S, T \in \mathcal{E}$, then there is an orthonormal basis of $V$ with respect to which every element of $\mathcal{E}$ has an upper-triangular matrix.

Suppose $\mathbf{F} = \mathbb {R}$ and $T \in \mathcal{L}(V)$. Prove that $T$ is self-adjoint if and only if all pairs of eigenvectors corresponding to distinct eigenvalues of $T$ are orthogonal and $V = E(\lambda_{1}, T) \oplus \cdots \oplus E(\lambda_{m}, T)$, where $\lambda_{1}, \ldots , \lambda_{m}$ denote the distinct eigenvalues of $T$.

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$. Prove that $T$ is normal if and only if all pairs of eigenvectors corresponding to distinct eigenvalues of $T$ are orthogonal and $V = E(\lambda_{1}, T) \oplus \cdots \oplus E(\lambda_{m}, T)$, where $\lambda_{1}, \ldots , \lambda_{m}$ denote the distinct eigenvalues of $T$.

Suppose $\mathbf{F} = \mathbb {C}$ and $\mathcal{E} \subseteq \mathcal{L}(V)$. Prove that there is an orthonormal basis of $V$ with respect to which every element of $\mathcal{E}$ has a diagonal matrix if and only if $S$ and $T$ are commuting normal operators for all $S, T \in \mathcal{E}$.

Suppose $\mathbf{F} = \mathbb {R}$ and $\mathcal{E} \subseteq \mathcal{L}(V)$. Prove that there is an orthonormal basis of $V$ with respect to which every element of $\mathcal{E}$ has a diagonal matrix if and only if $S$ and $T$ are commuting self-adjoint operators for all $S, T \in \mathcal{E}$.

Give an example of a real inner product space $V$, an operator $T \in \mathcal{L}(V)$, and real numbers $b, c$ with $b^{2}<4 c$ such that

is not invertible.

Suppose $T \in \mathcal{L}(V)$ is self-adjoint and $U$ is a subspace of $V$ that is invariant under $T$.

Suppose $T \in \mathcal{L}(V)$ is normal and $U$ is a subspace of $V$ that is invariant under $T$.

Suppose that $T$ is a self-adjoint operator on a finite-dimensional inner product space and that $2$ and $3$ are the only eigenvalues of $T$. Prove that

Give an example of an operator $T \in \mathcal{L}(\mathbb {C}^{3})$ such that $2$ and $3$ are the only eigenvalues of $T$ and $T^{2}-5 T+6 I \neq 0$.

Suppose $T \in \mathcal{L}(V)$ is self-adjoint, $\lambda \in \mathbf{F}$, and $\epsilon >0$. Suppose there exists $v \in V$ such that $\left\| v\right\| = 1$ and

Prove that $T$ has an eigenvalue $\lambda^{\prime }$ such that $\left|\lambda -\lambda^{\prime }\right|<\epsilon$.

Suppose $U$ is a finite-dimensional vector space and $T \in \mathcal{L}(U)$.

Suppose that $T \in \mathcal{L}(V)$ and there is an orthonormal basis $e_{1}, \ldots , e_{n}$ of $V$ consisting of eigenvectors of $T$, with corresponding eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$. Show that if $k \in \left\{ 1, \ldots , n\right\}$, then the pseudoinverse $T^{\dagger }$ satisfies the equation

Suppose $T \in \mathcal{L}(V)$. Prove that if both $T$ and $-T$ are positive operators, then $T = 0$.

Suppose $T \in \mathcal{L}\left(\mathbf{F}^{4}\right)$ is the operator whose matrix (with respect to the standard basis) is

Show that $T$ is an invertible positive operator.

Suppose $n$ is a positive integer and $T \in \mathcal{L}\left(\mathbf{F}^{n}\right)$ is the operator whose matrix (with respect to the standard basis) consists of all 1's. Show that $T$ is a positive operator.

Suppose $n$ is an integer with $n > 1$. Show that there exists an $n$-by-$n$ matrix $A$ such that all of the entries of $A$ are positive numbers and $A = A^{*}$, but the operator on $\mathbf{F}^{n}$ whose matrix (with respect to the standard basis) equals $A$ is not a positive operator.

Suppose $T \in \mathcal{L}(V)$ is self-adjoint. Prove that $T$ is a positive operator if and only if for every orthonormal basis $e_{1}, \ldots , e_{n}$ of $V$, all entries on the diagonal of $\mathcal{M}(T, (e_{1}, \ldots , e_{n}))$ are nonnegative numbers.

Prove that the sum of two positive operators on $V$ is a positive operator.

Suppose $S \in \mathcal{L}(V)$ is an invertible positive operator and $T \in \mathcal{L}(V)$ is a positive operator. Prove that $S+T$ is invertible.

Suppose $T \in \mathcal{L}(V)$. Prove that $T$ is a positive operator if and only if the pseudoinverse $T^{\dagger }$ is a positive operator.

Suppose $T \in \mathcal{L}(V)$ is a positive operator and $S \in \mathcal{L}(W, V)$. Prove that $S^{*} T S$ is a positive operator on $W$.

Suppose $T$ is a positive operator on $V$. Suppose $v, w \in V$ are such that

Prove that $v = w$.

Suppose $T$ is a positive operator on $V$ and $U$ is a subspace of $V$ invariant under $T$. Prove that $T|_{U} \in \mathcal{L}(U)$ is a positive operator on $U$.

Suppose $T \in \mathcal{L}(V)$ is a positive operator. Prove that $T^{k}$ is a positive operator for every positive integer $k$.

Suppose $T \in \mathcal{L}(V)$ is self-adjoint and $\alpha \in \mathbf{R}$.

Suppose $T$ is a positive operator on $V$ and $v_{1}, \ldots , v_{m} \in V$. Prove that

Suppose $T \in \mathcal{L}(V)$ is self-adjoint. Prove that there exist positive operators $A, B \in \mathcal{L}(V)$ such that

Suppose $T$ is a positive operator on $V$. Prove that

Suppose that $T \in \mathcal{L}(V)$ is a positive operator. Prove that there exists a polynomial $p$ with real coefficients such that $\sqrt{T} = p(T)$.

Suppose $S$ and $T$ are positive operators on $V$. Prove that $S T$ is a positive operator if and only if $S$ and $T$ commute.

Show that the identity operator on $\mathbf{F}^{2}$ has infinitely many self-adjoint square roots.

Suppose $T \in \mathcal{L}(V)$ and $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$. Prove that $T$ is a positive operator if and only if there exist $v_{1}, \ldots , v_{n} \in V$ such that

for all $j, k = 1, \ldots , n$.

Suppose $n$ is a positive integer. The $n$-by-$n$ Hilbert matrix is the $n$-by-$n$ matrix whose entry in row $j$, column $k$ is $\frac{1}{j+k-1}$. Suppose $T \in \mathcal{L}(V)$ is an operator whose matrix with respect to some orthonormal basis of $V$ is the $n$-by-$n$ Hilbert matrix. Prove that $T$ is a positive invertible operator.

Suppose $T \in \mathcal{L}(V)$ is a positive operator and $u \in V$ is such that $\left\| u\right\| = 1$ and $\left\| T u\right\| \geq \left\| T v\right\|$ for all $v \in V$ with $\left\| v\right\| = 1$. Show that $u$ is an eigenvector of $T$ corresponding to the largest eigenvalue of $T$.

For $T \in \mathcal{L}(V)$ and $u, v \in V$, define $\left\langle u, v \right\rangle_{T}$ by $\left\langle u, v \right\rangle_{T} = \left\langle T u, v \right\rangle$.

Suppose $S$ and $T$ are positive operators on $V$. Prove that

Let $T$ be the second derivative operator in Exercise 31(b) in Section 7A. Show that $-T$ is a positive operator.

Exercise 7A.31: fix a positive integer $n$. In the inner product space of continuous real-valued functions on $[-\pi , \pi ]$ with inner product $\left\langle f, g \right\rangle = \int_{-\pi }^{\pi } f g$, let $V = \operatorname {span}(1, \cos x, \cos 2 x, \ldots , \cos n x, \sin x, \sin 2 x, \ldots , \sin n x)$. Part (b) defines $T \in \mathcal{L}(V)$ by $T f = f''$.

Suppose $\dim V \geq 2$ and $S \in \mathcal{L}(V, W)$. Prove that $S$ is an isometry if and only if $S e_{1}, S e_{2}$ is an orthonormal list in $W$ for every orthonormal list $e_{1}, e_{2}$ of length two in $V$.

Suppose $T \in \mathcal{L}(V, W)$. Prove that $T$ is a scalar multiple of an isometry if and only if $T$ preserves orthogonality.

Suppose $\mathbf{F} = \mathbb {C}$ and $A, B \in \mathcal{L}(V)$ are self-adjoint. Show that $A+i B$ is unitary if and only if $A B = B A$ and $A^{2}+B^{2} = I$.

Suppose $S \in \mathcal{L}(V)$. Prove that the following are equivalent.

1. $S$ is a self-adjoint unitary operator.

2. $S = 2 P-I$ for some orthogonal projection $P$ on $V$.

3. There exists a subspace $U$ of $V$ such that $S u = u$ for every $u \in U$ and $S w = -w$ for every $w \in U^{\perp }$.

Suppose $T_{1}, T_{2}$ are both normal operators on $\mathbf{F}^{3}$ with $2, 5, 7$ as eigenvalues. Prove that there exists a unitary operator $S \in \mathcal{L}\left(\mathbf{F}^{3}\right)$ such that $T_{1} = S^{*} T_{2} S$.