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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Find a linear operator such that $\|T\| > \sup_{\|f\| \leq 1} \left|\langle Tf, f \rangle\right|$

Find a linear operator $T: \mathbb C^2 \to \mathbb C^2$ such that $$\|T\| > \sup_{\|f\| \leq 1} \left|\langle Tf, f \rangle\right|,$$ where $\langle \cdot, \cdot \rangle$ is the standard inner ...
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Show that $\pi(M)'' = \pi(M'')$

Let $M$ is a $*-$ subalgebra of $B(H)$. Let $\bar H$ denote the direct sum $\sum H_i$ where $\{H_i\}$ is a family of replicas of $H$. Define $$\pi :x\in B(H) \to \bar x \in B(\bar ...
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Computing Hermitian Conjugate for an Operator on a Function

The operator $\hat D$ is defined by $(\hat D f)(x) = \sqrt 2 f(2x)$. Show that $\hat D$ is a linear transformation, compute its hermitian conjugate and show it is unitary. Determine all eigenfunctions ...
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28 views

Inverse of a particular operator

I need help finding the inverse of the following operator. I am not sure about how to start. Any help would be hugely appreciated. Operator: $( I + \frac{\partial^2}{\partial x^2})$ Edit: I ...
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If $A$ is a $*-$ Banach algebra then $\bar A^{wot} = \bar A^{weak^*}$?

If $A$ is a $*-$ subalgebra of $B(H)$, then clearly $\bar A^{weak^*}\subset \bar A^{wot}$ (wot means weak operator topology). Also on every bounded subset of $A$, two topologies equal. Now my question ...
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Is this following bilinear form coercive?

First of all I want to mention that this is homework, so don't spoil it and reveal all the answer. just some guidenss :) Let $H$ be a Hilbert space, $T:H\rightarrow H$ a bounded linear operator for ...
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3answers
261 views

Spectrum of a nilpotent operator

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator such that $A^n=0$ for some $n\in \mathbb{N}$. Is the spectrum of $A$ finite, countable ?
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A convergence in norm topology

Let $H$ be a Hilbert space and $P_{n}\in B(H)$ be an increasing net of finite-rank projection which converge to the identity in the strong operator topology. Then, for any $T_{1}, T_{2}\in B(H)$, ...
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Trace of the exterior powers of linear operators

Given linear operators $K_1,\ldots,K_m$ on a Hilbert space $\mathcal H$, what can we say about the trace of their exterior product $Tr \,(K_1\wedge \cdots \wedge K_m)$ ? More precisely: 1) If we ...
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47 views

perturbation by orthogonal projection

Let $G$ be an operator with discrete spectrum on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$. Let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} = G+P$. My ...
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82 views

Are the invertible elements of a Banach algebra closed in the set of left-invertible elements?

Let $A$ be a unital Banach algebra. Denote by $\mathrm{Inv}(A)$ the invertible elements in $A$, and $\mathrm{Inv}_\ell(A)$ the left-invertible elements. That is, $a \in \mathrm{Inv}_\ell(A)$ if and ...
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Understanding dual spaces

My professor asked me to give him an example about dual spaces from our real life so I was thinking if we take the set of all our actions as the space X can the space of all our thoughts be the dual ...
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28 views

When is the Sturm-Liouville operator $ Lf=x^2f''+xf'$ positive

On the interval $[a,b]$ what conditions make the operator $L= (x^2)D^2 + xD$ positive? here $D$ is the differentiation operator.
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Two Body Schrodinger Equations

I have a question involving the eigenvalues of a two-body Schrodinger equation. Let $$H=-\frac{1}{2m}\Delta_{x_1}-\frac{1}{2m}\Delta_{x_2}+\frac{e^2}{|{{x_1}-{x_2}}|}$$ over the Hilbert space ...
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2answers
95 views

Self-adjoint operator restricted on a closed subspace

Let $A$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$ (possibly unbounded, densely defined with domain $\mathcal{D}(H)$) and let $S$ be a closed subspace of $\mathcal{H}$, ...
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1answer
102 views

Finding eigenfunctions and eigenvalues to Sturm-Liouville operator

I'm struggling to understand how to find the associated eigenfunctions and eigenvalues of a differential operator in Sturm-Liouville form. For instance, one question that I am trying to solve is the ...
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64 views

Partial Isometries: Introduction

Attention This question has been modified drastically. It is done so the answer below is still correct. It is done so to allow more specialized threads. Problem How do I deal with partial ...
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76 views

Example of a wot convergent net but not $\sigma -$ weak convergent

Let $B(H)$ be the space of bounded linear operators. Define the $\sigma-$ weak topology on it by seminorms $p_{h,k} (x)=|\sum_{n\geq 1}(xh_n,k_n)$ where $h=\{h_n\}\subset H ,~~ k=\{k_n\}\subset H $ ...
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49 views

Eigenvalues of an integral operator on $L^2[-1, 1]$

Find the eigenvalues of the integral operator $K: L^2[-1, 1] \to L^2[-1, 1]$ defined by $(Kx)(t) = \int_{-1}^1 (1 - 3t \tau)x(\tau) d\tau$. I began with the fact that eigenvalues must be values ...
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88 views

Show self-adjointness with eigenvalue expansion.

I was wondering if anybody here knows how to show that the negative Laplacian is self-adjoint on the 2 nd order Sobolev space of the two-sphere? I read that it is a rather cumbersome calculation, but ...
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43 views

Spectrum of unitary transform

Let $T: \operatorname{dom}(T) \rightarrow H$ be self-adjoint, then $U(T):=(T+i)(T-i)^{-1}$ is defined and unitary( this is clear to me). Furthermore, we have that $\sigma(U(T)):= \overline{\{ t; ...
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48 views

Diagonalisability of Self-Adjoint Operators for Non-Symmetric Metrics

Let $V$ be a finite dimensional vector space and $(\cdot,\cdot)$ a non-degenerate bilinear form. When $(\cdot,\cdot)$ is symmetric, every self-adjoint operator on $V$ is diagonalisable. What happens ...
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Convolution Operator and Integration Operator

I have some questions about the following two operators. A convolution operator $T$. If $k \in \mathcal L^1(\mathbb R)$, then $$f(x) \mapsto \int_{-\infty}^\infty k(x-y)f(y) dy: \mathcal L^2(\mathbb ...
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Møller Operators: Unitary Equivalence

Reference This is taken out of M. Reed and B. Simon, Scattering Theory. Problem Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the ...
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Møller Operators: Absolutely Continuous Subspaces [closed]

Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the Møller operators: $$\Omega^\pm(H,H_0):=\mathrm{s-lim}_{\tau\to\pm\infty}e^{i\tau ...
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54 views

Comparing weak and weak operator topology

We can compare topologies on $B(H)$. For instance, Sot topology is stronger than wot topology or $\sigma-$ weak topology is equivalent to weak* topology. I would like to compare wot topology and weak ...
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66 views

A question about local convexity of the weak operator topology

By definition, I know a locally convex space is a topological vector space whose topology is defined by a family of seminorms $\cal P$ such that $$\bigcap_{p\in{\cal P}}\{x\colon p(x)=0\}=\{0\}.$$ ...
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40 views

Closed Operators: Empty Spectrum

Are there operators on Hilbert space having empty spectrum? (Surely, for Banach spaces they do exists.) Necessarily, they must be closed and can't be normal.
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How does one diagonalise an operator that has exponential elements?

I asked this question before on the Physics StackExchange, but as one commenter noted I might have more luck here. So the question is: What is the diagonal form of the (density) operator $\hat\rho$, ...
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42 views

Show existence and uniqueness of integral equality with neumann-series

I want to show that for $$x(s)-\int_0^12rs\cdot x(r)dr=\sin(\pi s)$$ there exists exactly one solution $x \in C^0([0,1],\mathbb R)$.
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51 views

Positive operator has a positive spectrum?

Let $T : \operatorname{dom}(T) \rightarrow H $ be a positive self-adjoint operator, is it then true that $\sigma(T) \subset [0,\infty)$? This is something that sounds natural and I guess that it is ...
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Spectral Measures: Spectrum vs. Numerical Range

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{D}(N)\to\mathcal{H}$. The goal here is to prove: $$\langle\sigma(N)\rangle=\mathcal{W}(N)$$ By a previous result one has: ...
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Integro-differential operator

Good morning everybody, recently I came across the following question: is it possible to characterize (i.e. giving differential conditions which are necessary AND sufficient) the solutions of the ...
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Translational invariance and zero eigenvalue

Page 2 (506), line 18 of http://www-personal.umich.edu/~orosz/articles/NonlinScipublished.pdf says that "The presence of translational symmetry in the nonlinear equations gives rise to a relevant ...
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1answer
60 views

An inequality about operator norm

Let $H$ be a Hilbert space and $T\in B(H)$, with $T_{i}\rightarrow T$ in strong operator topology. Then can we prove that $\liminf_{i\rightarrow \infty}||T_{i}||\geq ||T||$ ?
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130 views

Classification of operators

I have a collection of questions about the limit point/circle concept and self-adjointness that are kind of connected, so I would like to ask them in a row. Apparently, an operator that is limit ...
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Confusion about the definition of self adjoint and formally self-adjoint

I have some confusion about the definition of self-adjoint operators and formally self-adjoint operators. Let me write down the background information. Let $H$ be a infinite dimensional complex ...
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Possible flaw in “proof” that a sum of two compact operators is compact

If X and Y are Banach spaces, and $A: X \to Y$, $B: X \to Y$ are both compact operators, then $A + B$ is compact. A + B is compact if and only if for every bounded sequence $\lbrace x_n \rbrace$ ...
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Proving that if $<Ax,x>=0$ for every $x$, then $A$ is the zero operator

I feel kind of dumb but I needed this little claim as a part of a proof I'm writing, and I figured out that I'd better just ask, since I could not find the correct algebraic manipulation needed in ...
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1answer
50 views

(Possible) application of Sarason interpolation theorem

This question is related to the following Wikipedia article on Nevanlinna–Pick interpolation. At the end it has been written as Pick–Nevanlinna interpolation was introduced into robust control by ...
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Does Hilbert–Schmidt theorem imply the space is separable?

The Hilbert–Schmidt theorem says a self-adjoint compact operator on a Hilbert space have a complete orthonormal set consisting of eigenvectors. Does that imply the space is separable?
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Why is the total time derivative of this partial space derivative zero?

A Lax pair for the Burgers equation $u_t+2 \, u \, u_x+ u_{xx} =0$ is, $$L = \partial_x +u \text{ and } M=-\partial_{xx} -2 \, u \, \partial_{x}$$ To get the resulting differential equation from the ...
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Compact operators on a Banach space $X$ are closed in the bounded operators on $X$. - Proof correction help

I am given a proof of the following statement (see below). Compact operators on a Banach space $X$ are closed in the bounded operators on $X$. I was told that there is an error in this proof - I ...
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Prove $Tx=(r_1x_1, r_2x_2, r_3x_3,…)$ is compact, $T:l^2\to l^2$, $r\in l^2$

Here is my question: Fix $r=(r_1,r_2,...)\in l^2$. Define $T:l^2\to l^2$ by $$Tx=(r_1x_1, r_2x_2, r_3x_3,...)$$ Prove that $T$ is compact. Here is what I have, input would be appreciated: Let ...
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Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$

Let $A$ be a bounded or unbounded selfadjoint linear operator on a complex Hilbert space $H$ with spectral representation $A=\int_{\sigma}\lambda \, dE(\lambda)$ given by the Spectral Theorem for ...
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59 views

Inequality between operator norm and Hilbert-Schmidt norm

I have seen the following inequality here but I don't know where I can find a proof for it. Could somebody give me a hint to understand it or guide me to a reference please? $\|AB\|_{HS} \leq ...
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Let L be a bounded linear operator on a Hilbert space H. Verify the following relationships: $null(L^*)=null(LL^*)$

Just started to learn about linear operator theory, and trying to understand adjoint operator. Here's a conceptual problem, can someone help me to clarify? Thanks Let L be a bounded linear operator ...
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Spectral Measures: Commuting Operators

The questions are given below!! Theorem Given a measure space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. Denote ...
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102 views

How to use the Spectral Theorem to Derive $L^{2}(\mathbb{R})$ Fourier Transform Theory

Without using Fourier transforms, how do I derive the spectral measure for $A=\frac{1}{i}\frac{d}{dt}$ on the domain $\mathcal{D}(A)$ consisting of absolutely continuous functions $f\in ...
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Why is this operator one-to-one

I am reading a textbook, and would like to ask a question about the proof. Here $S_p$ is the Schatten p class. My question is, in the proof, why is $A: X\to H$ is one-to-one? I actually don't ...