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Questions tagged [operator-theory]

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.

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13 views

Relation between spectrum and operator support

$f$ is a function in $C^{\infty}$ with compact support and $A$ is a self-adjoint operator. If $\mathrm{supp}(f)\cap\sigma(A)=\varnothing$, does this imply that $f(A)=0$? Does anyone have an ...
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1answer
37 views

Please Help with Question about Linear Operators

The Question: Let $X$ and $Y$ be Banach spaces and $T: X \rightarrow Y$ an injective bounded linear operator. Show that if $R(T)$ is closed in Y, then $T^{-1} : R(T) \rightarrow X $ is bounded. My ...
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1answer
10 views

Abelian/Finite Projections in Von Neumann Algebras

I know that provided a von Neumann Algebra acting on Hilbert $(\mathcal{M},\mathcal{H})$ a projection $e \in P(\mathcal{M})$ is said to be abelian if $e\mathcal{M} e$ is $\textbf{abelian}$ and the ...
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2answers
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On polar decomposition

In the paper "Isometries of non-commutative $L_p$-spaces" by Yeadon the author states the following: Let $H$ is separable Hilbert space, $\mathcal B(H)$ is $\ast-$algebra of all bounded linear ...
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Bound for the operator norm of a symmeric Markov kernel acting on $L^2$ in terms of the spectral gap

Let $\kappa$ be a symmetric Markov kernel on a probability space $(E,\mathcal E,\mu)$. We know that $$\kappa_0f:=\int\kappa(\;\cdot\;,{\rm d}y)f(y)\;\;\;\text{for }f\in L^2_0(\mu)$$ is a self-adjoint ...
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If $A$ is self-adjoint, does the formula $r(A)=\max_{\lambda\in\sigma(A)}|\lambda|$ for the spectral radius even hold in the non-complex setting?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$, $H$ be a $\mathbb K$-Hilbert space, $A\in\mathfrak L(H)$ and $\sigma(A)$ and $r(A)$ denote the spectrum and spectral radius of $A$, respectively. ...
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31 views

If $A$ is self-adjoint, then $\left\|A\right\|=\max_{\lambda\in\sigma(A)}|\lambda|$

Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be self-adjoint with $\left\|A\right\|_{\mathfrak L(H)}\le1$, $\sigma(A)$ and $r(A)$ denote the spectrum and spectral radius of $A$, ...
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1answer
46 views

Studying the off-diagonal case of this operator $f \mapsto \int_{B(0,1)} f(x-y)dy $

I am trying to show that the operator $T$ defined on Schwarz class functions (on $\Bbb R^n$) by, $$f \mapsto \int_{B(0,1)} f(x-y)dy $$ (where $B(0,1)=\{t \in \Bbb R^n : ||t|| <1\}$ ) , is not ...
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Book on Operator Theory

I am new to things like the Contraction Mapping and am looking for a book to further my understanding. Are there any recommendations? Also, I wonder if one needs to have a good understanding of ...
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1answer
23 views

$TT^*$ is unitary equivalent to $T^*T$.

Let $H$ be a complex Hilbert space and let $\mathcal{B}(H)$ denote the bounded linear operators from $H$ to itself. An operator is said to be normal if $TT^* = T^*T$. I would like to know which ...
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18 views

Operator norm of symmetric Markov kernel acting on $L^2$

Let $(S,\mathcal B)$ be a measurable space, $\pi$ be a Markov kernel on $(S,\mathcal B)$ and $\mu$ be a probability measure on $(S,\mathcal B)$ invariant with respect to $\pi$. Let $L^2_0(\mu):=\left\{...
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20 views

Strong continuity of the square root on the set of positive operators [on hold]

I was looking at an exercise in Reed-Simon (vol. 1) which asks to show that if $A_n$ converges strongly to some $A$, then so does $\sqrt{A_n}$ to $\sqrt{A}$. The setting being positive operators in $B(...
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1answer
58 views

If $A$ is self-adjoint, then $\left\|A\right\|=\sup_{x\in H\setminus\{0\}}\frac{\langle Ax,x\rangle}{\left\|x\right\|^2}$

Let $H$ be a $\mathbb R$-Hilbert space and $A\in\mathfrak L(H)$ be self-adjoint. I want to show that $$\left\|A\right\|_{\mathfrak L(H)}=\sup_{x\in H\setminus\{0\}}\frac{\langle Ax,x\rangle_H}{\left\|...
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Linear Operator in a Banach Space

I'm trying to get my head around linear operators and their usage with Banach spaces. Could someone help me understand how some properties relate to the following operator? We have Banach space $L^2$,...
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26 views

Spectrum of compact operator on an infinite dimensional Banach space

Let $X$ be an infinite dimensional Banach space (over $\mathbb{C}$) and let $T\colon X\to X$ be a compact operator. Let $\sigma(T)$ denote the spectrum of $T$ and let $\sigma_{\text{p}}(T)$ denote the ...
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25 views

comparision with rank one projection

Suppose $P$ is a rank one projection in $B(H)$,if we have $T\in B(H)$ such that $0\leq T\leq P$,does the following conclusion hold: There exists $\alpha \geq 0$ such that $T=\alpha P.$
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1answer
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Span of Density operators (Positive Semi-definite matrices of Trace one)

While reading basic-mathematics of quantum mechanics I came across a statement - "For every complex euclidean space $\cal X$ there exists spanning sets of the space $L({\cal X})$ consisting of ...
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1answer
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Eigenfunctions of composition operators acting on the space of entire functions.

Let $\lambda$ and $b$ be complex numbers with $\lambda$ nonzero. Let $\varphi(z)=\lambda z+b$ and define the composition operator $C_\varphi$ on the space of entire functions by $$C_\varphi f(z) = f\...
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Interchange of linear operators

While I was studying Signal Theory, my book says this about LSS (Linear and Stationary Systems): [a system is said to be stationary if $y(t-t_0) = \mathcal{T}[x(t - t_0)]$] Let $\mathcal{T}$ be the ...
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1answer
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the norm of positive elements in $C^*$-algebra

Suppose $x,y$ are two positive elements in a unital $C^*$-algebra,if $x\leq y$,then $\|x\| \leq \|y\|$. If $x\geq 0,y\geq 0$,$\|x\| \leq \|y\|$,can we conclude that $x\leq y$?
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1answer
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Determine the range of ${\rm i}\frac{\rm d}{{\rm d}t}$ on $\left\{f\in L^2([0,1]):f\text{ is absolutely continuous and }f(0)=f(1)=0\right\}$

Let $\operatorname{AC}(0,1)$ denote the space of absolutely continuous functions $[0,1]\to\mathbb C$, $$Af:={\rm i}f'\;\;\;\text{for }f\in\mathcal D(A):=\left\{f\in L^2([0,1];\mathbb C):f\in\...
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1answer
40 views

Exploring more from Equivalent operator norm as $|⟨𝐴𝑢,𝑣⟩|$

Exploring more from Equivalent operator norm as $|\langle Au,v\rangle|$. $A$ is a linear bounded operator, and $H$ is a Hilbert space. Let $P := \sup \{|\langle Au,u\rangle| : u \in H,\ \|u\|=1\},$ ...
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Determine the adjoint of ${\rm i}\frac{\rm d}{{\rm d}t}$ on different domains

Let $\mathbb K\in\{\mathbb C,\mathbb R\}$, $\operatorname{AC}(0,1)$ denote the space of absolutely continuous functions $[0,1]\to\mathbb K$, $$\begin{align}&\mathcal D(A_0):=\left\{f\in L^2([0,1];\...
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Right way to write down or use an operator?

Let $f:X\times Y \rightarrow \mathbb{R}$. I want to simplify an operator to denote the following functional transformation: $$f(x,y) - \int_{Y}f(x,y)g(y)\,dy, $$ where $g$ is the pdf of $Y$. So I ...
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1answer
40 views

On Hyperfinite ness

If $M\subset B(\mathcal{H})$ is hyperfinite type $\mathrm{II}_1$ factor, does it imply $PM$ is again hyperfinite type $\mathrm{II}_1$, where $P$ is a projection in $M'$.
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Book recommendation: Functional Analysis for Graphs

There is a active area of research on generalising matrix-based methods for graph theory to the infinite case with operators. Topics of interest are generalisations of spectral graph theory and graph ...
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1answer
21 views

Show that the symmetric operator $Sx:=(nx_n)_{n\in\mathbb N}$ on $\mathcal D(S):=\{x\in H:(nx_n)_{n\in\mathbb N}\in\ell^2\}$ is even self-adjoint

Let $H:=\ell^2(\mathbb N,\mathbb C)$ and $$Sx:=(nx_n)_{n\in\mathbb N}\;\;\;\text{for }x\in\mathcal D(S):=\left\{x\in H:(nx_n)_{n\in\mathbb N}\in H\right\}.$$ $S$ should be bijective and $T:=S^{-1}\in\...
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1answer
30 views

Inclusion of numerical range and translation and rotation invariance

Let $T\in\mathscr{B(\mathcal{H})}$ and $X\in M_n(\mathbb{C})$. Then prove that following two are equivalent: (i) $W(B\otimes X)\subseteq W(B\otimes T)$, for all $B\in M_n$ (ii) $W(C\otimes (aX+bI_n))...
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1answer
31 views

Is $T^{\ast\ast}\subseteq T^\ast$ obvious for a symmetric operator $T$?

Let $H$ be a $\mathbb C$-Hilbert space and $T$ be a densely-defined symmetric linear operator on $H$. Can we show that $T^{\ast\ast}\subseteq T^\ast$ without using the fact that $T$ is closable and ...
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1answer
32 views

Automorphism with no square root

Related to Every normal operator on a separable Hilbert space has a square root that commutes with it Does it exist an automorphism $f$ in a separable $\mathbb C$ Hilbert space, such that $f$ has no ...
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23 views

isomorphism between subspaces of Banach/Hilbert spaces

Let $X,Y$ be Banach/Hilbert spaces. Let $T:V\to W$ be an (linear) isomorphism, where $V$ is a subspace of $X$ and $W$ is a subspace of Y. Consider $T^*$,the adjoint of $T$. Which conditions $T$ ...
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1answer
48 views

Can we show that $T^{\ast\ast}\subseteq T^\ast$?

Let $H$ be a $\mathbb C$-Hilbert space and $T$ be a densely-defined linear operator on $H$. Can we show that $T^{\ast\ast}\subseteq T^\ast$? It's clear to me that the claim is true when $T$ is ...
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0answers
35 views

Right inverse with minimal spectral radius

Let $A$ be a right invertible operator acting on a Hilbert space $\mathcal{H} $. I am trying to find a right inverse of $A$ that has the minimal spectal radius possible, that is an operator $B$ such ...
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0answers
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Equivalent definition for a densely-defined operator to be symmetric

Let $H$ be a $\mathbb C$-Hilbert space, $T$ be a densely-defined linear operator on $H$ and $\Gamma(T)$ denote the graph of $T$. By definition, $T$ is symmetric if $T\subseteq T^\ast$ and self-adjoint ...
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1answer
25 views

Exponential of self-adjoint operator

Let $A$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$. If $u_A:=\sum_{n=0}^\infty \frac{i^nA^nu}{n!}$ is converge, $u_A=e^{iA}u$. I cannot prove this. I will appreciate your help ...
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1answer
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$(Fx)(t) = f(t)x(t)$ is unitary iff $|f(t)| = 1$ almost everywhere

The problem comes from Naylor's Linear Operator Theory Section 5.19 Problem 4 Let $I = [a, b]$ be a bounded interval and define $F : L_2(I) → L_2(I)$ by $(F x)(t) = f(t)x(t)$, where $f ∈ L_∞(I)$. ...
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Show that the adjoint of a certain multiplication operator is not densely-defined

Let $f:\mathbb R\to\mathbb C$ be bounded and Borel measurable, $\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$ and $$M_fg:=fg\;\;\;\text{for }g\in\mathcal D(M_f):=\left\{g\in L^2(\...
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Change of Variables in Infinitesimal Generator

I have a random process $\chi$, which has infinitesimal generator $$\mathcal{L} = A\bigg(\chi^2\frac{\partial^2}{\partial \chi^2}+\chi\frac{\partial}{\partial \chi}\bigg), \quad A\in\mathbb{R}.$$ I ...
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1answer
36 views

categorization of $C^*$ algebra

There are three types of Von Neumann algebras,namely,Type $I,II,III$ VNA .I wonder whether the $C^*$ algebra can be categorized completely?
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56 views

If $T$ is a densely-defined injective operator between Hilbert spaces with dense range, then $T^\ast$ is injective as well

Let $H_i$ be a $\mathbb C$-Hilbert space and $T$ be a densely-defined linear operator from $H_1$ to $H_2$. How can we show that if $T$ is injective and $\operatorname{im}T$ is dense, then $T^\ast$ ...
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1answer
11 views

If $U$ is an unitary operator, then $U(M^\perp)=U(M)^\perp$

Let $H_i$ be a $\mathbb C$-Hilbert space, $U$ be an unitary linear operator from $H_1$ to $H_2$ and $M\subseteq H_1$. How can we show that $U(M^\perp)=U(M)^\perp$? Clearly, if $x\in M^\perp$ and $y\...
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0answers
35 views

A cyclic vector for the Dirichlet Laplacian on interval.

While revisiting the notion of a cyclic vector for a self-adjoint operator (see, e.g., Schmuedgen, Unbounded Self-adjoint Operators on Hilbert space, Section 5.4), I got stuck when I tried to figure ...
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1answer
26 views

Existence and uniqueness of the adjoint of a linear operator between Hilbert spaces

Let $H_i$ be a $\mathbb C$-Hilbert space, $T$ be a linear operator from $H_1\to H_2$ and $T^\ast$ be a linear operator from $H_2$ to $H_1$ with $$\langle T^\ast y,x\rangle_{H_1}=\langle y,Tx\rangle_{...
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1answer
52 views

Approximation of selfadjoint matrix in $M_n(\mathbb{C})$

We first observe that by spectral theorem of a selfadjoint matrix, any selfadjoint matrix $A\in M_n(\mathbb{C})$ can be written as (up to unitarily equivalence) $A=A_{+}-A_{-}$ where $A_{+}, A_{-}\geq ...
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0answers
37 views

multiplicative operators

The following Functional Equation $$f(x+y+xy)=f(x)+f(y)+f(x)f(y)$$ is called Pompeiu Functional Equation. The solution to this equation is the Pompeiu Function given by $$f(x)=M(x+1)-1$$ where M is ...
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1answer
37 views

A tuple of Hilbert space operators

Let $(\mathcal{H}, \langle\cdot,\cdot\rangle)$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ is the algebra of all bounded linear operators on $\mathcal{H}$. For ${\bf T} = (T_1,\cdots,...
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2answers
16 views

Trace Class Operators and Compactness

According to many sources I've looked through, part of the definition of trace-class operators is that they be compact. What's the need for this caveat? Why not just look at all those operators $T$ ...
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0answers
19 views

Why does $p(x,\xi)$ satisfy the growth condition in the definition of a pseudo differential operator

I am following Shanahan's book about Atiyah Singer index theorem, on page 89, there is a confusion that troubles me for a long time, but I haven't find the details about it. As above stated. First, ...
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1answer
13 views

General formula for the following operator?

I am looking for a general formula for the following linear operator $T_n[f]$. $$T_0[f]=1$$ $$T_1[f]=\pi f(x)-\frac12$$ $$T_2[f]=2i\pi f'(x)-\pi f(x) +\frac16$$ etc. In general, $T_n[f]$ is equal ...
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1answer
35 views

How to show the explicit form of rotation operator?

Suppose $P_R(\alpha) f(x,y)=f(x\cos\alpha +y\sin\alpha, y\cos\alpha-x\sin\alpha)$ How to show that the operator $P_R(\alpha)$ has the form: $$ P_R(\alpha)=1+\sum_{n=1}^\infty \frac{1}{n!}(-i\alpha)^...