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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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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0answers
23 views

Lie group of differential operators

I have the following three partial differential operators $$A=y \frac{\partial}{\partial y}$$ $$B=y^{-1}(z\frac{\partial}{\partial z}+y\frac{\partial}{\partial y}+c-1)$$ $$C=y((1-z)\frac{\...
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1answer
29 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
44 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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52 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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18 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( \begin{array}...
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1answer
42 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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36 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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1answer
320 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 "$\forall a,b \in A, ab=0 \iff ba=0$" ...
4
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1answer
60 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 E\|X\...
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0answers
27 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
35 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
39 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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0answers
14 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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0answers
97 views

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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2answers
218 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
45 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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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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1answer
37 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
35 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
56 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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0answers
22 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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0answers
24 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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2answers
77 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
60 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 ...
2
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1answer
35 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 $a_2(...
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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 $L^*...
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0answers
69 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 $x=\sum_{j=...
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1answer
37 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
38 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 \right\...
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2answers
72 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 ...
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0answers
43 views

Proof compactness of adjoint operators

I am trying to understand a proof of the following statement: Given a complex B-space X and a compact operator $T:X\rightarrow X$, the adjoint operator $T^\ast:X^\ast \rightarrow X^\ast$ is compact as ...
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1answer
25 views

If $A$ is the Laplacian on $H^2(0,1)∩H_0^1(D)$, then the fractional power space $\mathfrak D(A^{r/2})=H_0^r(D)$ for all $r\in\mathbb R$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
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27 views

If $G$ is the Green's function of the Laplacian $A$ and $L$ is the integral operator with kernel $G$, then $L$ is the inverse of $A$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
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25 views

How do we compute the Green's function of the Laplacian?

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
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2answers
49 views

What is the proof that linear operators can be treated as variables?

I understand what a linear operator is, but I don't understand why you can just treat it as a variable.
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1answer
31 views

Bilinear Form in Definition of Adjoint Operator

I'm working through some material in Analysis Now by Pedersen, and I'm a little confused about how the adjoint to an operator is defined. Pederson defines it by: If $X$ and $Y$ are normed spaces ...
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3answers
42 views

Example of a Projection Operator in $\mathbb{R^3}$

I'm looking for an operator $\hat P$ in $\mathbb{R^3}$ such that $\hat P^2=\hat P$ that is also Hermitian
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1answer
29 views

Is a bounded, linear, nonnegative and symmetric operator with finite trace on a Hilbert space Hilbert-Schmidt?

Let $U=(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $Q$ be a bounded, linear, ...
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1answer
57 views

Approximate unit for a certain C*-algebra

Let $A$ be a C*-algebra and $p$ a projection in $A^{**}$. To prove $p$ is the smallest unit for $B: = \{a\in A; pap=a\}$, suppose $\{u_i\}$ is an approximate unit for $B$. It's easy to see $q: = w^*-\...
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1answer
50 views

Trace of multiplication operator on $L^2(\mathbb{T})$

Let $H=L^2(\mathbb{T})$, where $\mathbb{T}$ is the Torus. Consider a multiplication operator with a sufficiently nice function $f$. Is there somehow a formula like $$\mathrm{tr} M_f = C \int_{\mathbb{...
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1answer
23 views

Prove Linear Operator $(T_{w}f)(x) = |\dot{w}|^{\frac{1}{2}} f(w(x)) $ Is Unitary

Let the inner product be given by: $$\langle f,g \rangle = \int_{-\infty}^{\infty} f(x)g^{\ast}(x)dx$$ Then I want to show: $\langle T_{w}f, T_{w}g \rangle = \langle f,g \rangle$ So we have: $$\...
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1answer
32 views

what is the operator name that for positive value returns the same and for negatvie returns zero?

there is an operator that currently I see in mechanical engineering topics that for positive values returns the same quantity inside, and for negative values, returns zero , it's like <> but a bit ...
2
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1answer
35 views

Operator theory problem

Show that operator $ T : C([0,1]) \to C([0,1]) $, where $$ Ty(t) = \int_{0}^{1} |x-t|^{-1/2}y(x) \ dx. $$ Calculate the norm T. Thank for help.
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0answers
50 views

Show that $S_n \to S $(weakly) and $T_n \to T$ strongly implies $S_nT_n \to ST$ weakly

Let $X,Y,Z$ be Banach Spaces. Let $T_n,T \subset BL(X,Y), S_n,S \in BL(Y,Z)$. Show that a) $S_n \to S $(weakly) and $T_n \to T$ (strongly) implies $S_nT_n \to ST$ (weakly) b) $S_n \to S $(uniformly)...
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0answers
41 views

Is the Hankel matrix defined by $\ln(k)/k$ a bounded operator on $\ell^2$?

I call a sequence $(b_k)$ of positive real numbers a Hilbert kernel if there is a constant $C > 0$ such that $$ \sum_{k=1}^\infty\sum_{j=1}^\infty b_{k+j}c_kc_j\,\le\,C\sum_{k=1}^\infty c_k^2 $$ ...