Functional Calculus

Let $A$ be an operator in an $n$-dimensional space $L$ over the field $K$. Prove that the operators $1,A,A^2,\ldots ,A^n$ are linearly dependent.

Prove that the following properties of an operator $A$ on an $n$-dimensional space $L$ over the field $K=\mathbb {R}$ or $\mathbb {C}$ are equivalent:

1. the operators $1,A,A^2,\ldots ,A^{n-1}$ are linearly independent;

2. there is a vector $\xi \in L$ such that $\xi ,A\xi ,\ldots ,A^{n-1}\xi$ is a basis in $L$;

3. there is a vector $\xi \in L$ that is cyclic for $A$ (i.e., $L$ itself is the only subspace of $L$ containing $\xi$ and invariant under $A$). Operators $A$ having these properties are called regular.

Prove that a diagonal matrix determines a regular operator if and only if the elements on the diagonal are distinct.

Prove that the following properties of a matrix $A$ are equivalent:

1. $A$ defines a regular operator;

2. the minimal polynomial for $A$ coincides with the characteristic polynomial;

3. there is only one Jordan block for each eigenvalue of $A$.

Prove that the set of regular operators is open and dense in the set of all operators.

Let $R_n$ be the collection of $n\times n$ matrices of the form

Prove that

Prove that two regular operators $A$ and $B$ on an $n$-dimensional space $L$ over the field $K=\mathbb {R}$ or $\mathbb {C}$ are similar if and only if

Let

be a Jordan block of order $n$ with eigenvalue $\lambda$. Compute the matrices:

Let $\mathfrak A$ be an algebra over the field $K=\mathbb {C}$ or $\mathbb {R}$. An idempotent is defined to be an element $x\in \mathfrak A$ having the property $x^2=x$. The direct sum of algebras $\mathfrak A_1$ and $\mathfrak A_2$ is defined to be the linear space $\mathfrak A_1\oplus \mathfrak A_2$ with componentwise multiplication. Prove that the following properties of $\mathfrak A$ are equivalent:

1. $\mathfrak A$ is isomorphic to the direct sum of some (nonzero) algebras $\mathfrak A_1$ and $\mathfrak A_2$;

2. $\mathfrak A$ contains a nontrivial (different from $0$ and $1$) idempotent. Algebras not having these properties will be called irreducible.

Prove that every finite-dimensional algebra is a direct sum of irreducible algebras.

A numerical sequence $\left\{ a_n\right\}$ has the property that $0\leq a_{m+n}\leq a_m+a_n$ for all $m$ and $n$. Prove that $\lim_{n\to \infty }(a_n/n)$ exists and is equal to $\inf_n(a_n/n)$.

Let $A$ be an operator in an $n$-dimensional linear space $L$ over a field $K$. Denote by $\mathfrak A(A)$ the algebra over $K$ generated by $1$ (the identity operator) and $A$. Prove that $\dim \mathfrak A(A)\leq n$.

Let $K=\mathbb {C}$. Prove that the algebra $\mathfrak A(A)$ is irreducible if and only if $A$ has a unique eigenvalue.

Let $S$ be some set of operators in a linear space $L$. Let $S'$ be the collection of operators in $L$ that commute with all the operators in $S$. For what operators $A$ does the equality $\mathfrak A(A)'=\mathfrak A(A)$ hold?

Prove that every polynomial in the coefficients of the matrix $A$ that is invariant under the similarity transformations $A\mapsto CAC^{-1}$ is a polynomial in $\operatorname {tr} A,\operatorname {tr} A^2,\ldots ,\operatorname {tr} A^n$.

Let $A$ and $B$ be $2\times 2$ matrices. Prove that every polynomial in the coefficients of $A$ and $B$ that is invariant under the substitutions $A\to CAC^{-1}$, $B\to CBC^{-1}$ has the form $P(\operatorname {tr} A,\operatorname {tr} B,\operatorname {tr} A^2,\operatorname {tr} B^2,\operatorname {tr} AB)$, where $P$ is some uniquely determined polynomial in five variables.

Let $A$ and $B$ be $n\times n$ matrices. Prove that the algebra of those polynomials in the coefficients of $A$ and $B$ that are invariant under the substitutions $A\to CAC^{-1}$, $B\to CBC^{-1}$ contains no fewer than $n^2+1$ generators.

In the space of $2n\times 2n$ matrices find a subspace of dimension $1+n^2$ consisting of pairwise commuting matrices.

Let $A$ be an operator in an $n$-dimensional space with a single eigenvalue $\lambda$. Prove that $f(A)=\sum_{k=0}^{n-1}[f^{(k)}(\lambda )/k!](A-\lambda \cdot 1)^k$ for any function $f$ that is $n-1$ times differentiable at the point $\lambda$.

Let $A$ be an operator in an $n$-dimensional space with distinct eigenvalues $\lambda_1,\ldots ,\lambda_n$. Prove the formula

Suppose that the operator $A$ has eigenvalues $\lambda_1,\ldots ,\lambda_n$ with multiplicities $m_1,\ldots ,m_n$. Prove the formula

and find the explicit form of the operators $B_{jk}$.

Let $K$ be the collection of all positive (see Problem 643) operators with trace $1$ on a finite-dimensional Hilbert space $H$. Prove that $K$ is a convex compact set, and find the extreme points of $K$.

Let $A$ be the operator of multiplication by a continuous real function $a(x)$ on the space $L^{2}\left({0,1}\right)$. Prove that $A$ is selfadjoint, and find $\sigma (A)$.

Find the spectrum of the operator $A$ acting in $L^{2}\left({0,1}\right)$ according to the formula $Af(x)=a(x)f(x)$, where $a\in L^{\infty }\left({0,1}\right)$.

Let $f\in L^{1}\left({\mathbb {R},dx}\right)$. Find the spectrum of the convolution operator $S(f)$ on the space $L^{2}\left({\mathbb {R},dx}\right)$.

Let $f\in L^{1}\left({\mathrm T,dt}\right)$. Find the spectrum of the operator $S(f)$ of convolution with $f$ on the space $L^{2}\left({\mathrm T,dt}\right)$.

Prove that the spectrum of a unitary operator $U$ lies on the unit circle.

Let $A$ be a selfadjoint operator. Prove that the operator $(A+\lambda 1)(A+\overline{\lambda }1)^{-1}$ is unitary for nonreal $\lambda$.

Suppose that the operator $(A-i1)$ is invertible and the operator $(A+i1)(A-i1)^{-1}$ is unitary. Prove that $A$ is selfadjoint.

It is known that the operator $U$ is unitary, and $U-1$ is invertible. Prove that the operator $A=i(U+1)(U-1)^{-1}$ is selfadjoint.

Compute the spectral radius of the Volterra operator $A$ on $L^{2}\left({0,1}\right)$ defined by the formula

Compute explicitly the resolvent of the Volterra operator in Problem 641.

An operator $A$ on a Hilbert space $H$ is said to be positive if $\left\langle Ax, x \right\rangle \geq 0$ for all $x\in H$, $x\ne 0$. In this case write $A\gg 0$. Prove that a positive operator $A$ satisfies the formula

Let $A$ be a selfadjoint operator such that $a\cdot 1\ll A\ll b\cdot 1$, and $p(x)$ a nonnegative polynomial on $[a,b]$. Prove that $p(A)\gg 0$.

Prove that the mapping $p\mapsto p(A)$ is continuous relative to the norm in $C[a,b]$ if $a\cdot 1\ll A\ll b\cdot 1$.

Let $A$ be a bounded selfadjoint operator. Prove that $U(t)=e^{itA}$ is a unitary operator for all $t\in \mathbb {R}$ and that

Prove that under the conditions of Problem 646 the operator-valued function $U(t)$ is differentiable and $U'(t)=iAU(t)=iU(t)A$.

Prove that every operator-valued function $U(t)$ that is norm-continuous and satisfies the equations $U(t)U(s)=U(t+s)$, $U(t)^*=U(-t)$ has the form indicated in Problem 646.

Find the polar decomposition of the operator $A$ of multiplication by a function $a\in L^{\infty }\left({X,\mu }\right)$ on $L^{2}\left({X,\mu }\right)$.

Find the polar decomposition of the one-sided shift operator on $l_2(\mathbb {C})$.

Let $A$ and $B$ be commuting operators, and $A=RU$ the polar decomposition of $A$.

Suppose that $A\gg B\gg 0$ and that $B$ is invertible. Prove that $A$ is invertible and $A^{-1}\ll B^{-1}$.

Let $T$ be the shift operator in $l_2(\mathbb {Z})$ ($T\left\{ x_n\right\} =\left\{ x_{n+1}\right\}$). Prove that there exists a unique selfadjoint operator $A$ with the following properties:

1. $T=e^{iA}$;

2. $\left\| A\right\| \leq \pi$.

Let $H_1$ and $H_2$ be subspaces of $H$, and $P_1$ and $P_2$ the corresponding orthogonal projections. Prove that $\lim_{n\to \infty }(P_1P_2)^n$ exists and is equal to the orthogonal projection onto $H_1\cap H_2$.

Let $A$ be the operator on $L^{2}\left({[(0,\infty ),dx]}\right)$ given by the formula $Af(x)=\int_0^\infty [f(y)/(x+y)]\, dy$. Prove that $A$ commutes with the dilation operators $L(a):f(x)\mapsto f(ax)$.

In the notation of Theorem 7 in Ch. V prove that $\tau (\Gamma_A)^\perp$ is the graph of some operator if and only if $D_A$ is dense in $H$.

Here $\Gamma_A=\left\{ x\oplus Ax:x\in D_A\right\} \subset H\oplus H$, and $\tau (x\oplus y)=(-y)\oplus x$. Theorem 7 states that $\Gamma_{A^*}=\tau (\Gamma_A)^\perp$.

Suppose that the operators $A$ and $A^*$ are densely defined (i.e., $D_A$ and $D_{A^*}$ are dense in $H$). Prove that $(A^*)^*$ coincides with the closure of $A$.

Prove that each of the following conditions is equivalent to the essential selfadjointness of an operator $A$:

1. $A^*$ is selfadjoint;

2. $\bar A=A^*$.

In which of the following cases is the operator $A=i\, d/dx$ in $H=L^{2}\left({0,1}\right)$ symmetric, essentially selfadjoint, or selfadjoint:

Determine whether the Laplace operator $\Delta =\partial^2/\partial x_1^2+\partial^2/\partial x_2^2$ is symmetric in $L^{2}\left({\mathbb {R}^{2},dx}\right)$ when:

Show that every symmetric operator $A$ for which $D_A=H$ is bounded.

Let $A$ be a selfadjoint operator.

Let $U$ be a unitary operator for which $\ker (U-1)=\left\{ 0\right\}$. Prove that the operator $A=i(U+1)(U-1)^{-1}$ with domain $D_A=\operatorname {im}(U-1)$ is selfadjoint.

Compute the operator $A$ in Problem 663 if $U$ is the operator of shift by $1$ in $l_2(\mathbb {Z})$.

Suppose that $H=l_2(\mathbb {C})$, $D_A$ consists of all the finitely nonzero sequences with zero sum, and $A$ is the operator given by the matrix $A=(a_{jk})$, where $a_{jk}=i\operatorname {sign}\left(j-k\right)$.

Let $A_i$, $i=1,2$, be the operator of multiplication by $x$ in $L^{2}\left({\mathbb {R},dx}\right)$ with domains $D_{A_i}$. Suppose that $A_1$ and $A_2$ are essentially selfadjoint. Is it possible that the subspaces $D_{A_1}$ and $D_{A_2}$ have zero intersection?

Suppose that the operator $A$ is densely defined and positive (see Problem 643). Prove that $A$ is essentially selfadjoint if and only if $\ker (A^*+1)=\left\{ 0\right\}$.

Prove that for any closed densely defined operator $A$ the operator $T=A^*A+1$ with domain $D_\tau =\left\{ x\in D_A;\ Ax\in D_{A^*}\right\}$ is selfadjoint.

Suppose that $H_1$ and $H_2$ are Hilbert spaces, and $H_1\otimes H_2$ their Hilbert tensor product. Prove that if $A_1$ and $A_2$ are selfadjoint operators in $H_1$ and $H_2$, respectively, then the operator $A_1\otimes 1+1\otimes A_2$, with domain $D_{A_1}\otimes D_{A_2}$, is essentially selfadjoint in $H$.

Prove that the spectrum of a selfadjoint operator lies on the real axis.

Prove that a selfadjoint operator $A$ does not have a symmetric extension different from $A$ itself.

Let $A$ be a symmetric operator. Prove that $(A+i)(A-i)^{-1}$ extends to an isometry $U$ from $\overline{\operatorname {im}(A-i)}$ to $\overline{\operatorname {im}(A+i)}$.

Let $A$ be a closed symmetric operator. Prove that the space $\operatorname {im}(A+i1)$ is closed in $H$.

Prove that a closed symmetric operator $A$ admits a selfadjoint extension if and only if $\dim \ker (A^*-i1)=\dim \ker (A^*+i1)$.