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
986 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
513 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
3answers
395 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
336 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
123 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
248 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 ...
7
votes
0answers
224 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
472 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
2answers
7k views

Norm of a symmetric matrix equals spectral radius

How do I prove that the norm of a matrix equals the absolutely largest eigenvalue of the matrix? This is the precise question: Let $A$ be a symmetric $n \times n$ matrix. Consider $A$ as an ...
6
votes
1answer
606 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
932 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
3answers
3k views

Norm of integral operator in $L_2$

What is the norm of integral operators $A$ in $L_2(0,1)$? $Ax(t)=\int_0^tx(s)ds$
6
votes
2answers
195 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
3answers
269 views

Relation between $K$-theories

I apologize in advance if this question is too vague/general. I am curious to know how all of the different $K$-theories are related (algebraic $K$-theory, topological $K$-theory, twisted $K$-theory, ...
6
votes
1answer
1k views

When to use Closed Graph Theorem vs. Uniform Boundedness Theorem?

I run in to problem that I often know is solvable with either the Closed Graph Theorem or Uniform Boundedness Theorem. I seem to mix up the solutions. Are there any hints on when to use which? Or can ...
6
votes
1answer
287 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
414 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 ...
6
votes
1answer
489 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
206 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
619 views

Sum of Closed Operators Closable?

Let $A$ and $B$ be closed operators on a (separable complex) Hilbert space with dense domains $D(A)$ and $D(B)$ respecitvely. Then, we may define the operator $A+B$ on $D(A)\cap D(B)$. In general, ...
6
votes
2answers
67 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
2k views

What is 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 ...
6
votes
1answer
90 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
1answer
174 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 ...
6
votes
2answers
607 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
93 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
309 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
241 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
662 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
865 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 ...
6
votes
1answer
39 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
106 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
317 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 ...
6
votes
1answer
242 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
80 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 ...
6
votes
1answer
109 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? ...
6
votes
1answer
107 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
264 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
137 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
762 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
706 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} ...
6
votes
1answer
114 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
44 views

Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$

Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$, looking at $C_{(2)}[-1,1]$, with $L_2$ norm. I tried to look at a general polynomial $\sum_{i=0}^{98} ...
6
votes
2answers
79 views

The subgroup of $GL_{n}(\mathbb{R})$ inside $B(1, 1)$ is $\lbrace I \rbrace$

Let $G \subset GL_{n}(\mathbb{R})$ be a subgroup such that $G \subset B$ where $$B = \lbrace M \in \mathcal{M}_{n}(\mathbb{R}), ||M-I|| < 1\rbrace $$ Let $g \in G$. I'm asked to show: The ...
6
votes
1answer
89 views

Is the Neumann series a compact operator?

Let $X$ be an infinite dimensional Banach space and $A:X\to X$ be a compact operator with the operator norm $\|A\|<1$. Then $I-A$ is invertible and the Neumann series $$ S_N = \sum_{k=0}^N A^k $$ ...
6
votes
1answer
200 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
6
votes
1answer
593 views

When solving a linear differential equation by factoring the operator, how does one guarantee no solutions are lost?

I think the best way to make this question clear is with an example. Lets say we want to solve the differential equation $(\Delta^2 - \lambda^4)\phi=0,$ calculations are greatly simplified if we ...