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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Graph of linear operator

I have a few questions regarding the graph of an operator. Consider the operator $T:X \rightarrow Y$ between Banach spaces $X,Y$. Assume that $T$ is a linear operator which is (weak, weak)-continuous, ...
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Positive invertable element of a C*- algebra

The following is Theorem 2.2.5 of Murphy's C*-algebras and operator theory: Let $A$ be an unital C*-algebra and $a,b$ are positive invertable elements, if $a\leq b$, then $0\leq b^{-1}\leq a^{-1}$. ...
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15 views

Green-Operator for Sturm-Liouville Differential equation compact on Sobolev space?

Let $g$ be Green's Function for a Sturm-Liouville differential equation. Is the operator $G: H_{0}^{1}(0,1) \rightarrow H_{0}^{1}(0,1)$ defined by $(Gf)(x) := \int_{0}^{1} g(x,y)f(y) dy, \quad f \in ...
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Visual notion of tangential gradient

Before I begin, this question is related to personal reading and is not in any way connected to an assessment/assignment. I am struggling to visualise the tangential gradient. As I understand it, the ...
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53 views

Is this equality true or it is not necessarily true?

Let $A$ and $B$ are two factor von neumann algebras that act on two infinite dimensional Hilbert spaces H and K respectively. Let $\Phi:A\longrightarrow B$ is an additive bijective map with some other ...
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Product of two positive compact, self adjoint operators

If we have two positive compact , self adjoint operators; $A$, $B$. Is the product $AB$ a positive operator?
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19 views

Spectral Measures: Subspace Characterization

Disclaimer This thread is related to: Spectral Measures: Subspace Decomposition It is meant to record. See: Answer own Question It is written as question. Have fun! :) Question Given a Hilbert ...
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2answers
28 views

solvability condition for differential operator

While reading the research article I came across following derivation, given a self-adjoint operator, \begin{eqnarray} L = \frac{d^2}{dx^2} + f(x) \end{eqnarray} \begin{eqnarray} L\psi_1(x) ...
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1answer
17 views

An example of an unbounded non-orthogonal projection in a Hilbert space

What is an example of an unbounded non-orthogonal projection in a Hilbert spaces? Does it exist? A non-orthogonal projection is an idempotent operator: $T^2=T$. So the question is: can such an ...
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9 views

Power series of bounded linear operators

If $f$ is a complex analytic function, one can define a matrix function $F$ using the Taylor series of $f$ by $$ F(A) = f(0) + f'(0)\cdot x + f''(0)\cdot \frac{A^2}{2!} + \cdots $$ If the radius of ...
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Laplace transform and Fourier transform of kernel

Suppose $p_t(x)=\frac1{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}$ is the Gaussian density. $p_t(x)$ is also the Green kernel of the heat equation in 1D: $(\partial_t-\frac12\Delta)u=0$. The Fourier transform ...
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38 views

Spectrum of a Self-Adjoint Operator is Real

Preparing for an exam in functional analysis, I'm trying to show that for a self-adjoint operator $A$, $\sigma(A) \subset \mathbb{R}$. I came across the following proof in the book (or rather, lecture ...
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31 views

Proving monotonicity of continuous linear functional

Hi I am interested in resolving the following problem from the bottom of page 147 from a paper I am revising: Given a function $$a: \Omega \times \mathbb{R} \times \mathbb{R}^{N} \rightarrow ...
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30 views

Does Linearity imply Commutativity? [on hold]

If I have two linear operation X and Y , could I conclude that for X + Y = Y + X ?
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30 views

Find the eigenvalues and eigenvectors of T in V

Let $\mathbf{V}$ be the linear span of the functions 1, cos x, sin x. Let the operator T on V be given by the rule $T y(x)= y(x+\pi/4)$. Find the eigenvalues and eigenvectors of T in V. I'm not sure ...
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121 views

If a linear operator has an adjoint operator, it is bounded

This is a question I'm struggling with for a while: Let $H$ be a Hilber space. Let $T,S: H\rightarrow H$ be linear operators (not neccessarily bounded) such that for every $x,y\in H$: $\langle ...
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23 views

When is $\|\phi(|T|)\|=\|\phi(T)\|$ for $T\in B(H)$?

If $T\in B(H)$, $H$ a Hilbert space, then it has a polar decomposition $T=V|T|$, where $|T|=(T^*T)^{1/2}$ and and $V$ is a partial isometry. Let $\phi:B(H)\to B(H)$ be ucp (unital completely ...
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What conditions must an operator meet, to have only real eigenvalues?

Given the problem $Lu = \lambda u$, what properties must $L$ have, for all its eigenvalues to be real? An answer in the context of (partial) differential equations would be appreciated.
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Does any chaotic operator $T$ in infinite-dimensional Hilbert space is an isomorphism?

Suppose that any operator is an "Isomorphism" if it is mapping a hard equation to easier algebraic equations. Can I take this as a necessary criteria to judge that: "Any chaotic operator $T$ in ...
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1answer
19 views

Showing that the trace of a positive operator is independent of orthonormal base [closed]

let $T$ be a positive operator on a separable Hilbert space. let $\{e_n\}$ be an orthonormal base for the space, and suppose the trace of $T$ is finite, i.e. $$tr(T)= \sum_{n=1}^{\infty}(Te_n,e_n) ...
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42 views

$\sigma$-weak topology versus weak operator topology

The reference text for this question is: Pedersen, Analysis Now, GTM 118. The $\sigma$-weak topology on $B(H)$ (the bounded linear operators on a Hilbert space $H$) is the weak$^*$-topology on ...
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28 views

Polar decomposition

Every $x\in B(H)$ has a representation such as $x= u|x|$ (by polar decomposition) where $u$ is a partial isometry. In a paper, the author claims $$u = strong - \lim_{\epsilon\to 0} ...
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31 views

Ultraweak closed left ideal of a von Neumann algebra

The following is a proposition of Takesaki's Operator Theory: My questions are: 1- He claims for two sided ideal $\cal m$, $e \in M\cap M'$. While I think for $\sigma -$ weakly closed two sided ...
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find eigenvalue of a integral operator.

I'm having trouble finding all the eigenvalues of the following operator: $Tx(t)=\int_{-1}^1(1-3tz)x(z)dz$ In the examples I've seen, there's no systematic way that I learnt to do this. Can someone ...
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36 views

Why are integral and differential operators commutative?

For instance, let's assume a constant 3D surface over time $S$. $$ \frac{d}{dt}\iint_S \mathbf B \cdot \mathbf{ds} \quad=\quad \iint_S\frac{\partial \mathbf B}{\partial t}\cdot \mathbf{ds} $$ Why ...
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Operator which is invariant in terms of left shift

Let $l:l^{\infty}(\mathbb R)\rightarrow \mathbb R$ be a linear map( where $l^{\infty}$ is the set of bounded sequences) such that the following holds: (i) $l(Tx)=l(x)$, where T is the left shift ...
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2answers
26 views

Unbounded operator

Assume you have an operator $T : \operatorname{dom(T)}\rightarrow H$. Now we also know that $ran(T)$ is finite-dimensional. Does this imply that $T$ is bounded?( So is $T$ a bounded map $T \in ...
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Standard examples of operator

In a text I am reading it says that we can consider an operator $A: X \rightarrow X^{*}$ (where $X := W^{1,p}(\Omega)$) which is defined as $$Au = -\text{div}(a(x,u,\nabla u))$$ where $a: \Omega ...
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1answer
40 views

Computing a Projection Valued Measure

I've recently begun learning about Projection Valued Measure and I'm a little confused. I understand that a Projection Valued Measure is a family of orthogonal projections $P(\Lambda)$ indexed by the ...
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26 views

Differential operator and its inverse

suppose we have a differential operator $D$ in terms of a variable $x$ and its inverse is denoted by $D^{-1}$ , then is it possible that $DD^{-1}=\delta (x)$ or $D^{-1}D=\delta (x)$? If so, then what ...
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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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1answer
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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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24 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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$(\lim x_n)y = \lim (x_n y)$ when $x_n, y \in B(H)$

Please give me a hint to prove $(\lim x_n)y = \lim (x_n y)$ when $x_n, y \in B(H)$ for every n. Thanks in advance.
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1answer
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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
142 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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44 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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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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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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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
31 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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1answer
29 views

Partial Isometries: Characterizations

Any partial isometry satisfies: $$\Omega\Omega^*\Omega=\Omega$$ From this, one derives projections: $$\Omega^*\Omega,\Omega\Omega^*$$ Conversely, given projections: $$\Omega^*\Omega,\Omega\Omega^*$$ ...
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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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36 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 ...