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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Boundedness and norm of a linear operator

Consider the linear operator $T : C[-\pi,\pi] \to \mathbb{R}$ defined by $$ Tf := \int_{-\pi}^{\pi} f(t)\sin(t)\phantom{.}dt $$ Show that $T$ is bounded and find its norm $\|T\|$. Consider ...
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Generalization of matrix inversion lemma

I am looking for an operator version of matrix inversion lemma. To be specific, does the identity also hold for operators defined on general (infinitely dimensional) Hilbert space, possibly with ...
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Meaning of non-degenerate representation in $C^*$-algebras

A representation of a $C^*$-algebra, $A$, is a pair $(H,\pi)$ where $H$ is a Hilbert space and $\pi$ is a *-homomorphism from $A$ to $B(H)$. A representation is non-degenerate if $\{\pi(a)h:a\in A, ...
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open sets in a Banach space are locally connected

I'm reading a proof of the following theorem in operator algebra and I don't understand the first sentence: Would anybody show me why the following statement is true? Let $X$ be a connected ...
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Exponent of an Exponential Operator

There is a problem in my textbook that asks me to prove the following: For a bounded operator $A$ on a Hilbert space, prove that: $$(e^A)^n = e^{An} $$ for any natural number, $n$. However upon ...
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What is this operator topology?

Let $X$ be a separable Banach space with (norm $1$) Schauder-Basis $\{e_n\}_{n\in\mathbb N}$. Denote for $x\in X$ with $|\cdot|_x$ the seminorm on $\mathcal L(X)$ given by $|A|_x = \|A x\|$. Consider: ...
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Invariant subspaces for this linear extension of operators

Let $(e_k)$ be a total orthonormal sequence in a separable Hilbert space $H$ and let $ T: H\to H$ be defined at $e_k$ by $T(e_k)=e_{k+1}$ , $(k=1,2,\cdots)$ and then linearly and ...
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Determining eigenvalues of a differential or integral operator in Matlab?

Say I have a differential operator such as $L[\phi] = \frac{\partial \phi}{\partial x}$, or $L = \Delta \phi$, or an integral operator such as $L[\phi](x) = \int_{\partial D} \log(x - y) \phi(y) ...
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Spectrum of the derivative operator: What's wrong in my argument?

Consider the Banach space $X=C[0,1]$ of continuous functions $f:[0,1]\to\mathbb{R}$ equipped with the supremum norm. If we consider the following unbounded operator $A$ defined on its domain ...
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Proving an isometric dilation of a non unitary operator on Hilbert space implies infinite dimensional space involving matrices

I have been given this exercise in my Operator theory class dealing with operators on Hilbert spaces, which reads as follows: Let H be a Hilbert space. We are to prove, in two distinct ways, that ...
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Density of sets whose image is dense.

This is probably easy, but I can't think of an answer. Assume $X$ is a Banach space and $A$ is a (not assumed closed) subspace of $X$. Let $T:X \to X$ be a bounded operator, which is also injective. ...
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Spectral Measures: Integrability

I really need this as tool for other threads! Given a Hilbert space $\mathcal{H}$. Also a Borel space $\Omega$. Consider a spectral measure: $$E:\mathcal{B}(\Omega)\to\mathcal{P}(\mathcal{H}):\quad ...
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39 views

Eigenvectors Operators and Unilateral Shifts

We showed that a (non-zero) compact self-adjoint operator on a Hilbert space always has an eigenvector. Let $V:l^2(\mathbb{N})\to l^2(\mathbb{N})$ be the unilateral shift, the unique bounded operator ...
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45 views

Unitary Operators & Compact Self-Adjoint Operators

Let $U$ be a bounded operator on a Hilbert space. Show that the following are equivalent: I. $U$ is surjective and $\|Uv\|=\|v\|$ for all $v\in H$; II. $U$ is surjective and $\langle ...
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Bounded Operators: Topological Dual

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider the bounded operators: $$\mathcal{B}(\mathcal{H},\mathcal{K}):=\{T:\mathcal{H}\to\mathcal{K}:\|T\|<\infty\}$$ Regard the linear ...
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Bochner Integral of Positive Operators

So we have two function spaces (real or complex) X and Y (think $L^p$) and we say that a linear operator $P : X \to Y$ is positive if $f \geq 0$ implies $P(f) \geq 0$. I'm curious when a general ...
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why is the order of operations (for multiplications and division) giving different result?

Firstly sorry if this is tagged incorrectly or blindly obvious but it is confusing me a lot and I am not sure what category it would fall under. I have a particularly scenario where I am using the ...
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What is “analytic vector for closed operator”?

I need the defenition of "analytic vector of closed operator that acts on Hilbert space". I cant find it in google and in my textbooks (Khelemsky "Lectures And Exercises on Functional Analysis"), I ...
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Question on operator theory classes of operators on Hilbert spaces

I was recently tackled by this in my class on operator theory dealing with operators on Hilbert spaces: We are to find and prove the inclusion relations between the classes of operators: ...
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In which sense is composition a tensor product

Let $\Phi\colon U\to V$ and $\Psi\colon V \to W$ be linear operators, and consider their composition $$ \Psi\circ \Phi $$ The operation, $$\circ:\mathcal{L}(U,V)\times\mathcal{L}(V,W)\to ...
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Are quantum operators associative?

Let H be the Hamiltonian representing the total energy of the potential and kinetic component. But because all classical dynamical variables can be written as a function of position, x, and momentum, ...
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Does $\langle u,Tu \rangle=0$ imply that $T=0$? [duplicate]

I have this simple question : for an operator $T$ in a complex Hilbert space we have: $\langle u,Tu \rangle =0$ for all $u$ in this Hilbert space. So does this imply that $T=0$? If yes, how to ...
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Let $H$ be a self-adjoint operator with domain $D(H)$ in a Hilbert space. How to prove $He^{itH}u=e^{itH}Hu$?

Let $H$ be a self-adjoint operator with domain $D(H)$ in a Hilbert space and the function $e_t:\mathbb{R}\rightarrow \mathbb{C}$ continuous and bounded, defined by $e_t(x)=e^{itx}$. I want to know ...
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Why is $(\sqrt{P})^2=P$ where $P$ is a positive operator on a Hilbert space?

The following is a proposition regarding positive operators on a Hilbert space in Douglas's Banach Algebra Techniques in Operator Theory: Corollary 4.32 is as the following: I understand that the ...
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28 views

Closedness and continuity in infinite dimensional spaces

I cannot understand why the operator $A=d/dx: D(A)(\subset C[a,b])\to C[a,b]$ is closed when the domain $D(A)$ is chosen to be $C^1[a,b]$ while we know that we can converge to a non-differentiable ...
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43 views

Property of bounded linear transformation between Hilbert spaces

I've asked a question on related question in a previous thread, but I wanted to ask a follow up question. If a bounded linear transformation $T: X \to Y$ where $X$ and $Y$ are Hilbert spaces has ...
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14 views

$C^\ast$ condition implies $B^\ast$ condition

By $C^\ast$ condition I understand $\|A^\ast A\|=\|A^\ast\|\|A\|$ and for $B^\ast$, $\|A^\ast A\|=\|A\|^2$. I know these conditions are equivalent even NOT assuming the involution is isometric, but I ...
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28 views

Is range of completely continuous of bounded set finite dimensional set?

Define of completely continuous operator : $L$ is continuous operator and map bounded set to relatively compact set , then $L$ is completely continuous operator. Let $\Omega$ is a bounded set of ...
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63 views

Morphism: Unitization

Given C*-Algebras $\mathcal{A}$ and $\mathcal{B}$. (Possibly unital!) Morphisms are contractive: $$\varphi:\mathcal{A}\to\mathcal{B}:\quad\|\varphi\|\leq1$$ (Possibly nonunital!) How to apply ...
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Proving a variant of closed range theorem on Hilbert space

I've been working on closed range theorem. There are a lot of materials on general Banach spaces, but not much on Hilbert spaces, so I was wondering if I could get some help. I'm trying to prove the ...
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Lie group of differential operators

I have the following three partial differential operators $$A=y \frac{\partial}{\partial y}$$ $$B=y^{-1}(z\frac{\partial}{\partial z}+y\frac{\partial}{\partial y}+c-1)$$ ...
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Is any bounded operator weakly sequentially closed?

I have a theorem telling me that some property holds for operators that are bounded and weakly sequentially closed. Somehow, I have in mind that boundedness actually implies the weakly sequentially ...
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43 views

Linear operator satisfy $\operatorname{dim}(ran(A)) \le \operatorname{dim}(ker(A)^{\perp})$

Is it true that for a general bounded linear operator we have $\operatorname{dim}(ran(A)) \le \operatorname{dim}(ker(A)^{\perp})$? On finite-dimensional spaces we clearly have equality from matrix ...
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Do compact convergence topology and w*-topology coincide on the Pontryagin dual group of a LCA group.

Given a locally compact group $G$, consider its Pontryagin dual $G^{'}$. Do compact convergence topology and weak*-topology on $G^{'}$ coincide?. When we talk about weak*-topology on $G^{'}$, it ...
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Is this Hermitian matrix an example of an unbounded self adjoint operator?

I am trying to learn what an unbounded self adjoint operator is. Therefore I am asking if the following matrix $A$ is an example of an unbounded self adjoint operator: $$\LARGE A= \left( ...
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Hardy space on the upper plane

Recently,I need to study something about Hardy space. However, many books only contain Hardy space on the unit disk. Is there any book having detailed description about Hardy space on the upper plane ...
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Unbounded linear operator with bounded restriction

Given that a linear operator $T:X\rightarrow Y$, where $X$ and $Y$ are both Banach spaces, $D$ a dense subspace of $X$, if we know that the restriction of $T$ to $D$, say, $S=T|_{D}$ is bounded, then ...
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Is this a characterization of commutative $C^{*}$-algebras

Assume that $A$ is a $C^{*}$-algebra such that $\forall a,b \in A, ab=0 \iff ba=0$. Is $A$ necessarily a commutative algebra? In particular does "$\forall a,b \in A, ab=0 \iff ba=0$" ...
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Understanding the definition of the covariance operator

Let $\mathbb H$ be an arbitrary separable Hilbert space. The covariance operator $C:\mathbb H\to\mathbb H$ between two $\mathbb H$-valued zero mean random elements $X$ and $Y$ with $\operatorname ...
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Does spectral decomposition exist for non-self-adjoint operators?

In theory, if a linear operator $P$ in a Hilbert space $H$ is self-adjoint, we can decompose it as $Pu=\sum_i \lambda_i <\phi_i,u>\phi_i$, where $\phi_i$ is the eigenfunction of $P$. And we can ...
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How to prove Mellin transform on $L^2[0,1]$ is unitary?

Let $\{Im (s)\lt 0\}=\{s\in \mathbb{C}\mid Im(s)\lt 0\}$, and $H^2(\{Im (s)\lt0\})$ is the Hardy space on $\{Im (s)\lt 0\}$. I know a classical theorem of Paley and Wiener Fourier transform ...
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37 views

How to find the eigenfunctions of a differential operator.

Consider a linear differential operator $$L=\frac{d^2}{dx^2}.$$ How would one determine that the normalised eigenfunctions of $L$ are $$\phi_n(x)=\sqrt{2}\sin{(n\pi x)}?$$
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what does that operator means between 2 numbers?

(I couldn't write the operator in the title) It's ∨ as in a ∨ b = 839 e.g. I know this operator from boolean logic but I was surprised to find it in arithmetic. It was a question to find a and b to ...
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closability of $n$-th power of paranormal operator

It it well-known, that there exists closable paranormal operator $A$ such that $\overline{A}$ is not paranormal [1] and if $B$ is paranormal then $B^n$ is also paranormal [2]. Is there any example of ...
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A nilpotent element of an algebra which does not lie in the span of commutator elements.

What is an example of a $C^{*}$ algebra such that the span of nilpotent elements is not a sub vector space of the span of commutator elements. Obviously any such $C^{*}$ algebra would be a ...
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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 ...
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Completely continuous map is not homotopy with antipodal map

Define of completely continuous operator : $L$ is continuous operator and map bounded set to relatively compact set , then $L$ is completely continuous operator. Now, $E$ is a infinity dimensional ...
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4 views

Understanding certain symbols in “Non-Positive Partial Transpose Sub spaces Can be as large as any Entangled Subspace”

This is a link to a paper entitled "Non-Positive Partial Transpose Sub spaces Can be as large as any Entangled Subspace" by Nathaniel Johnston. I have issues understanding the notation used in this ...
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Does an essentially self-adjoint operator have the same kernel as its closure?

Let $H$ be a Hilbert space and let $A : D(A) \subset H \to H$ be an essentially self-adjoint operator. Let $\overline A$ be the unique self-adjoint extension of $A$. Question: Is it true that ...
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definition of block diagonal operator on a hilbert space

I 'm stuck with the definition of block diagonal operators on hilbert spaces. Def.: A bounded linear operator $T$ on a hilbert space $H$ is called block diagonal if there exists an increasing ...