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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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### 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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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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### 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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### 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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### 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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### 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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### $TT^*$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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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### $(Fx)(t) = f(t)x(t)$ 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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### 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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### 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, ...
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 ...
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)^...