Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

learn more… | top users | synonyms

7
votes
1answer
2k views

The difference between an isometric operator and a unitary operator on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is the identity operator) What is the difference between ...
7
votes
1answer
179 views

How to find interesting operators for a quantum system?

How can we find "interesting" operators for a quantum mechanical system? I can think of the following method: Given some system with an associated Hilbert space $V$ and Hamiltonian $H:V\rightarrow V$,...
7
votes
1answer
284 views

Trace of a differential operator

Given the differential operator: $$A=\exp(-\beta H)$$ where $$H=\frac{1}{2}\left( -\frac{d^2}{dx^2}+x^2 \right)$$ and $\beta\gt 0$ How can I get the trace of this operator? Thanks in advance.
7
votes
1answer
79 views

The ideal generated by a non-compact operator

I wanted to find a quick proof of the following well-known fact. Since I couldn't easily find a reference, I provide a proof below. Let $H$ be a separable Hilbert space, and $J\subset B(H)$ be a (...
7
votes
3answers
733 views

Prove $p^2=p$ and $qp=0$

I am not really aware what's going on in this question. I appreciate your help. Let $U$ be a vector space over a field $F$ and $p, q: U \rightarrow U$ linear maps. Assume $p+q = \text{id}_U$ and $pq=...
7
votes
1answer
82 views

A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm diag}(0,0,\cdots,0,1,1,1,1,\cdots)$$...
7
votes
1answer
210 views

True/False: Self-adjoint compact operator

Let $H$ be a hilbert space and $T$ a compact self-adjoint operator on it. T is also injective on a dense subspace $U \subset H$ and we also have that $T(H) \subset U$. Now I am asked whether it is ...
7
votes
1answer
371 views

Approximating a Hilbert-Schmidt operator

Let $H$ be a separable Hilbert space. Recall that a bounded operator $A : H \to H$ is said to be Hilbert-Schmidt if $$\|A\|_{HS}^2 := \sum_{i=1}^\infty \|A e_i\|^2 < \infty$$ where $\{e_i\}_{i=1}^\...
7
votes
1answer
1k views

Hilbert Schmidt integral operator

Hilbert-Schmidt Integral operators are usually defined from $H=L_2[a,b]$ into $H=L_2[a,b]$ as $$(Tf)(x) = \int_a^b K(x,y)f(y) dy,$$ provided that $K(x,y)$ is a Hilbert Schmidt kernel, namely $$\...
7
votes
2answers
517 views

On the isometry between bounded linear operators and the dual of nuclear linear operators

Let $H$ be a separable Hilbert space. Let $(e_i)_i$ be an orthonormal basis. For any bounded linear map $T$ we write, whenever possible $$\operatorname{tr} T := \sum_{i}^{\infty} \langle T e_i, e_i \...
7
votes
1answer
552 views

Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?

Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem Does the following generalization of that fact also hold? Theorem: ...
7
votes
3answers
398 views

Nonexistence of a cyclic vector for a representation on $\ell^2(\mathbb{Z})$

Let $S$ be the bilateral shift on $\ell^2(\mathbb{Z})$ and let $T = S + S^*$. I want to show that there is no cyclic vector for the representation of $T$ on $\ell^2(\mathbb{Z})$ i.e. $\forall x\in \...
7
votes
1answer
338 views

When are two commuting linear operators functions of each other

I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up. If we formally consider the integral operators $$...
7
votes
0answers
127 views

When does analytic in the operator norm imply analytic in the trace class norm?

Consider $U$ a nice compact region in $\mathbb{C}$ with boundary $\Gamma$. Let $S_1$ b the ideal of trace class operators on a separable complex Hilbert space $H$. We will let $\|\cdot \|$ be the ...
7
votes
1answer
260 views

Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a densely-...
7
votes
0answers
231 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
7
votes
0answers
477 views

Fixed point: linear operators

I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad. Consider a space $X$ ...
6
votes
1answer
610 views

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $. Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
6
votes
2answers
1k views

Prove that this integral operator is compact

Let $X,Y=L^2(0,1)$, $k\in C^0([0,1]^2)$. Define $$ K:X\to Y,\,\,\,\,\,Kf(x):=\int_0^1k(x,y)f(y)dy\,\,\,\,\forall\, f\in L^2(0,1). $$ I have to show that $K$ is compact. My idea is to prove that $K$ ...
6
votes
2answers
218 views

Why is $A$ a compact operator?

Let $X$ be a compact space and let $\mu$ be a positive Borel measure on X. Let $T\in \mathscr{B}(L^p(\mu),C(X))$ where $1\lt p \lt \infty$. Show that if $A:L^p(\mu)\rightarrow L^p(\mu)$ defined by $...
6
votes
1answer
1k views

Linear transformations in infinite dimensional vector spaces

If we look at an $n$ - dimensional vector space $V$ and a linear transformation \begin{equation} T : V \to V, \quad x \mapsto Tx \quad \forall \, x \in V \end{equation} then given a choice of basis ...
6
votes
2answers
351 views

For a multiplication operator $M_f$ on $L^2$ with $f\geq 0$, is $SM_fS^{*}$ positive?

I have the following problem. Let $\Omega \subset R^n$ have finite measure, let $H = L^2(\Omega)$ and let $S: H \to H$ be a bounded linear operator. Then it is well known that $P = SS^*$ is a positive ...
6
votes
1answer
290 views

boundedness of an operator

Define $T: L^2(\mathbb{R})\to L^2(\mathbb{R})$ by $(Tf)(x)=\int_{\mathbb{R}}\frac{f(y)}{1+|x|+|y|}dy$. Is this operator bounded? If it is, then is it also compact? I got stuck in simply applying ...
6
votes
2answers
418 views

Unitisation of $C^{*}$-algebras via double centralizers

In most of the books I read about $C^{*}$-algebras, the author usually embeds the algebra, say, $A$, as an ideal of $B(A)$, the algebra of bounded linear operators on $A$, by identifying $a$ and $M_a$,...
6
votes
1answer
560 views

Compact Operators: Weak Convergence [duplicate]

Problem Given Banach spaces $X$ and $Y$. Consider a compact operator $C\in\mathcal{C}(X,Y)$. Then weak convergence is turned into strong convergence: $$x_n\rightharpoonup x\implies Cx_n\to Cx$$ I'd ...
6
votes
3answers
207 views

Bounded operators with prescribed range - part II

This is a continuation of the question bellow, in a more particular case. Bounded operators with prescribed range If $X$ is a separable Banach space and $Y$ is a closed, infinite dimensional ...
6
votes
1answer
622 views

Fourier transform as a Gelfand transform

One question came to my mind while looking at the proof of Gelfand-Naimark theorem. Is Fourier transform a kind of Gelfand transform? Are there any other well-known transforms which are so?
6
votes
2answers
77 views

$P+Q-PQ$ is a projection if and only if $PQ=QP$.

Let $\mathcal H$ is a Hilbert space and $P,Q:\mathcal H \to \mathcal H$ are projections. I want to show that $P+Q-PQ$ is a projection if and only if $PQ=QP$. If $PQ=QP$ clearly $P+Q-PQ$ is a ...
6
votes
1answer
91 views

Can we say $TT^{*}=T^{2}$ implies $T=T^{*}$?

Let $A$ be a $C^{*}$-algebra, Can we say $TT^{*}=T^{2}$ implies $T^{*}=T$? for $T\in A$ I am looking for a counterexample! Thanks
6
votes
2answers
642 views

Double dual of the space $C[0,1]$

The second dual or double dual of the space of all continuous functions on $[0,1]$, $C[0,1]$ is von Neumann algebra. Can anyone help me identifying this space?
6
votes
3answers
95 views

Why is $\partial_z\partial_{\bar z}=\frac14\left(\partial_r^2+\frac1r\partial_r+\frac1{r^2}\partial_{\theta}^2\right)$?

I have to show the identity I wrote in the title: it should be $\partial_z\partial_{\bar z}=\frac14\left(\partial_r^2+\frac1r\partial_r+\frac1{r^2}\partial_{\theta}^2\right)$ but some computation ...
6
votes
1answer
318 views

What is the relationship between spectral resolution and spectral measure?

In Kadison and Ringrose's book "FUNDAMENTALS OF THE THEORY OF OPERATOR ALGEBRAS", the author gives the following theorem. Theorem: If $A$ is a self-adjoint operator acting on a Hilbert space $\...
6
votes
1answer
244 views

Matrix Representation of Trace Class Operators

Suppose we have a separable Hilbert space (thus with a countable basis) and that represent an operator in matrix form, i.e: $A: H \rightarrow H $$$x \;\rightarrow \sum_{j \in \mathbb{N}}\left(\sum_{k ...
6
votes
1answer
670 views

Symmetric Square Root of Symmetric Invertible Matrix

I am trying to find out if for any symmetric (Not necessarily self-adjoint), invertible matrix $A$ over $\mathbb{C}$, there is a square root of the matrix that is also symmetric. I was able to figure ...
6
votes
1answer
901 views

If $A_n$ is a sequence of positive bounded linear operators converging in norm to $A$ on a Hilbert Space, show $\sqrt{A_n}\to\sqrt{A}$ in norm.

If $A_n$ is a sequence of positive bounded linear operators converging in norm to $A$ on a Hilbert Space, show $\sqrt{A_n}\to\sqrt{A}$ in norm. I can show that $A$ would be positive and thus have a ...
6
votes
1answer
131 views

A Marcinkiewicz approach

The problem was to prove the following that the operator $$Tf(x)=\int_{\mathbb{R}^N}\frac{f(y)}{|x-y|^\alpha}dy$$ Is continuous from $$L^1 \to \ L_\mathrm{Weak}^{p}$$ where $0<\alpha<N$ and $p=...
6
votes
1answer
40 views

Convergence property of positive operator on $L^\infty$

Given a $\sigma$-additive measure space $(S,\Sigma,\mu)$ and a linear operator $$ U: L^\infty \to L^\infty $$ where $L^\infty$ is the space of essentially bounded measurable functions. Assume it is ...
6
votes
1answer
109 views

Show that $T$ is continuous

I have a question about how to response to: Given a Banach space X and $T: X \rightarrow X^{*}$ a linear operator such that $\langle Tx,x\rangle \geq 0$ for all $x \in X$. Show that $T$ is ...
6
votes
1answer
323 views

Weak* operator topology and finite rank operators

We will say that ${T_i}\subset B(X,Y^*)$ converges to $T$ in W*-operator topology if $T_i(x)\rightarrow T(x)$ in W*-topology of $Y^*$( $\forall y\in Y \langle T_i(x),y\rangle \rightarrow \langle T(x),...
6
votes
1answer
248 views

Is there a nonnormal operator with spectrum strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space. Definition : An operator $A \in B(H)$ is normal if $AA^{*} = A^{*}A$. Definition : The spectrum $\sigma(A)$ of $A \in B(H)$, is the set ...
6
votes
1answer
81 views

Is the adjoint operation WOT-WOT continuous?

This is a well-known property of the Hilbert-space adjoint operator that it is WOT continuous. Is a similar true for Banach spaces? That is, for a given Banach space $X$ is the operation $\varphi\...
6
votes
1answer
302 views

Relation between noncommutative geometry and functional analysis

Recently I came across the subject of noncommutative geometry via my interest in functional analysis. My very little exposure to this subject gives me a sense that part of it is built on the theory of ...
6
votes
1answer
110 views

local convexity of $L_p$ spaces

wiki says The spaces $L_p([0, 1])$ for $0 < p < 1$ are equipped with the F-norm they are not locally convex, since the only convex neighborhood of zero is the whole space Why is this so? http://...
6
votes
1answer
111 views

Unital nonabelian banach algebra where the only closed ideals are $\{0\}$ and $A$

This is a problem in exercise one of Murphy's book Find an example of a nonabelian unital Banach algebra $A$, where the only closed ideals are $\{0\}$ and $A$. But does such an algebra exist at ...
6
votes
1answer
118 views

Can $xy$ and $yx$ lie in different connected components of the group of invertible elements of an algebra?

What is an example of a Banach or $C^{*}$ algebra $A$ which has two invertible elements $x, y$ such that $xy$ can not be connected to $yx$ in $G(A)$, the space of invertible elements of $A$. A ...
6
votes
1answer
305 views

Definition of resolvent set

I'm having trouble understanding some subtlety of definition of resolvent set for given bounded operator A everywhere defined on some Hilbert space. Book I use (and many other sources) give the ...
6
votes
1answer
144 views

Compactness of the Volterra opelator

The Volterra operator is given as \begin{eqnarray} (Vf)(x)=\int_0^xK(x,y)f(y)\,{\rm d}y. \end{eqnarray} By the Arzelà–Ascoli theorem, $V\colon C^0[0,1]\rightarrow C^0[0,1]$ is compact operator. But, ...
6
votes
2answers
800 views

Matrix Representation of Operators in Infinite Dimensional (Separable) Hilbert Spaces

Suppose we have a separable Hilbert space (thus with a countable basis) and that we to represent an operator in matrix form, i.e: $$A: H \rightarrow H \\ \; \; \; \; \; \;x \;\rightarrow \sum_{j \in \...
6
votes
1answer
725 views

Every Hilbert-Schmidt is an integral operator?

Let $(X,\mu)$ be a $\sigma$-finite measure space. If $K\in\mathcal{L}^2(X\times X,\mu\times\mu)$ then the map $A_K:\mathcal{L}^2(X,\mu)\to\mathcal{L}^2(X,\mu)$ defined by\begin{equation} A_Kf(x)=\...
6
votes
1answer
113 views

K theory of finite dimenional Banach algebras

Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose $K_{0}$-group is a non trivial finite group?(I ...