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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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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1answer
21 views

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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1answer
16 views

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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2answers
45 views

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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12 views

Boundedness of a dirichlet form

Suppose $\mu$ is a finite measure on some space $\Omega$ (can simply be the Lebesgue measure on $[0, 1]$ or something like it). Let $S_1$, $S_2$ be two densely defined, symmetric, nonnegative ...
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83 views

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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18 views

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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2answers
68 views

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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1answer
32 views

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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34 views

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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1answer
25 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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1answer
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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13 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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1answer
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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2answers
60 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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2answers
72 views

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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21 views

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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1answer
26 views

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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1answer
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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37 views

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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17 views

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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1answer
37 views

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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31 views

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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297 views

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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1answer
57 views

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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24 views

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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1answer
17 views

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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1answer
33 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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26 views

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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13 views

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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194 views

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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1answer
43 views

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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3 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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1answer
33 views

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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1answer
29 views

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 ...
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2answers
43 views

What is the $C^*$-algebra generated by a normal operator?

The following is the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I don't find the definition for the $C^*$-algebra generated by a normal operator in the book. ...
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19 views

Norms under Conjugation by Projection Opertaros

I was reading about equivalent forms of the Kadison Singer Problem, and while looking at the Feichtinger Conjecture, I came across the claim that, for a projection operator $P$ and a self-adjoint ...
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1answer
21 views

Does pointwise nilpotency imply global nilpotency?

Is there a compact Haussdorf space $X$ and $C^{*}$ algebra $A$ with a continuous map $f:X\to A$, such that $f(x)\in A$ is a nilpotent element, $\forall x \in X$, but $f$ is not a ...
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23 views

Equivalence of Definitions of Strong Operator Topology

I have a couple questions about how we define the strong operator topology on $\mathscr{B} (H)$ that I'm hoping someone can help me with. First, I thought that the strong operator topology was the ...
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3answers
72 views

Understanding bounded linear operators

The definition of a bounded linear operator is a linear transformation $T$ between two normed vectors spaces $X$ and $Y$ such that the ratio of the norm of $T(v)$ to that of $v$ is bounded by the same ...
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1answer
50 views

Does elementwise matrix inequality extend to norms?

The elements of $A$ and $B$ are non-negative and $A_{ij} \leq B_{ij} \; \forall \; i,j$. Is it true that $\Vert A \Vert_p \leq \Vert B \Vert_p$ ? The norm is the operator norm induced by the usual ...
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26 views

Show that the operator is NOT symmetric.

Show that the Sturm-Liouville operator $L$ in $L^2([a,b])$ given by $$L=\frac{1}{r(x)}\left(DpD+q\right)$$ is not symmetric. I'm assuming $p=p(x)>0$ and $q=q(x)\geq 0$, as described by the problem ...
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1answer
33 views

Show that $L$ is formally self-adjoint.

Consider the differential operator $$L=e^xD^2+e^xD,\;\;D=\frac{d}{dx},\;0\leq x\leq1,$$ $$u^\prime(0)=0,\;\;\; u(1)=0.$$ Show that $L$ is formally self-adjoint. I just don't really know how to start ...
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30 views

self-adjoint operators and linear dependence

Let $L$ be a self-adjoint differential operator given by $L=\frac{d}{dx}\left(a_2\frac{d}{dx}\right)+a_0$. If $u_1$ and $u_2$ are two solutions of $Lu=0$ and $J(u_1,u_2)=0$ for some $x$ for which ...
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24 views

concomitant and self-adjoint operator

If $Lu = u^{\prime\prime}+\omega^2u$, show that $L$ is formally self-adjoint and the concomitant is $J(u,v)=vu^\prime-uv^\prime$. Moreover, if $u$ is a solution of $Lu=0$ and $v$ is a solution of ...
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57 views

Projection operator in Banach space is continuous

Let $(X,||\cdot ||)$ be a Banach space with a Schauder basis, i.e. there exist $e_j \in X$, $j\in \mathbb{N}$, s.t. $||e_j||=1$ for all $j$ and every $x\in X$ can be uniquely represented as ...
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1answer
34 views

Operator/Matrix inequality

Let $A,B$ be non-negative matrices, such that $0\leq B\leq 1$. Is it true that $BAB\leq A$? (meant in the quadratic form sense) $A,B$ do not need to commute in general.
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1answer
35 views

Dual space ($X^{*}$) and $X^{**}$

According to my lecture notes (we're using Folland' Real Analysis textbook), if $X$ is a normed vector space, then $L(X,Y) = \left\lbrace \text{all bounded linear operators T} : X \rightarrow Y ...
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2answers
65 views

Given $S \in B(Y^{*}, X^{*})$, does there exist $T\in B(X,Y)$ such that $S=T^{*}$?

Let $X, Y$ be Banach spaces, $S \in B(Y^{*}, X^{*})$. Does such operator $T \in B(X, Y)$ exist so that $T^{*}=S$? I suppose that the answer should be - no. Are there any hints that might help in ...