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Questions tagged [operator-theory]

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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Continuity of $\textrm{argmin}$ set-valued mapping

Let $X \subset \mathbb{R}^n$ be a finite set and $\Phi : X \mapsto 2^{X}$ be a set-valued mapping defined as follows: $\Phi(y) := \underset{x \in X}{\textrm{argmin}} \; L(x, y)$. I'm trying to ...
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Operator Semigroups Simplified

How do I explain Operator Semigroups, in particular, positive operator semigroups to someone who hasn't studied math beyond high school? I just want to give a vague idea/analogy to someone to let ...
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2answers
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Show that $A:\ell^2(\mathbb N)\to\ell^2(\mathbb N)$ where $A(e_n)=\lambda_ne_n$ is bounded.

Let $C\subset\mathbb C$ be closed. As $\mathbb C$ is separable then so too is the subset $C$. This means that there exists a countable subset $\{\lambda_n:n\in\mathbb N\}\subset C$ dense in $C$. In ...
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Show that $(f_n)_n$ is relatively compact in $L^p$ space

Let $I=[0,1]$, $Q=I\times I$ and $(u_n),(v_n)$ bounded sequences in $L^2(I)$. Assume $x\mapsto u_n(x), x\mapsto v(x)$ are continuous and monotone non decreasing on $I$ for all $n\in\mathbb{N}$; define\...
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Linear operator image subspace chain

How to prove the proposition: $A: V \to V$ is a linear operator on a finite-dimensional vector space $V $, if $Im(A^p)=Im(A^{p+1})$,then $Im(A^{p+1})=Im(A^{p+2})$ The "Kernel" version is simple but I ...
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1answer
37 views

$l_2(S)$is a hilbert space where S is a subset

Let S be a non-empty set and $l_2(S)$ be the set of all complex functions $f$ defined on $S$ with the following two properties: $(1) \{s:f(s)\ne0\}$ is empty or countable. $(2) \sum{|f(s)|}^2 <...
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3answers
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Norm and compactness of the Operator $(Tu)(x)=\alpha(x)u'(x), u\in Y, x\in I$

Let $I=[0,1]$ and call $X$ the Banach space $C(I)$, endowed with the uniform norm. Introduce $Y=\{u\in X, u$ diffentiable on $I$ with $u'\in X\}$ and set $||u||_Y=||u||_\infty+||u'||_\infty, u\in Y, ...
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If an operator $T$ satisfy a property, then $\|Tx\|=c\|x\|$

Let $(E,\langle\cdot\;,\;\cdot\rangle)$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$. Assume that $T\in \mathcal{L}(E)$ and satisfy the following property (P)...
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Showing convexity of a set in $\mathbb{C}^k$

Let $\mathcal{H}$ be an infinite dimensional separable Hilbert Space and $T\in\mathscr{B(\mathcal{H})}$. Suppose $$\mu_k(T):=\left\{ \begin{pmatrix} \lambda_1 \\ \lambda_2 \\ \vdots \\ ...
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Boundedness of a linear operator

Let $X$ be a real normed linear space of all real sequences which are eventually zero with the 'sup' norm and $T:X \to X$ be a bijective linear operator defined by $$T(x_1,x_2,x_3,....)=\left(x_1,\...
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1answer
26 views

Norm of Fourier series

I am reading the proof of the statement that no non-zero multiplication operator on $L^2([0,1])$ is compact in this post. And I would like to address it as a seperate post as I am only curious about ...
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1answer
13 views

Examples of closed operators with nonempty resolvent set. [on hold]

I am looking for some (non trivial) examples of unbounded closed operators defined in Banach or Hilbert spaces and having nonempty resolvent set, where the resolvent set of a closed operator is: $ \...
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Is $i: (C^1, ||·||_{W^{1,2}}) → (C^0, ||·||_∞)$ a linear, continuous, compact map?

Consider the map $$i: (C^1[0,1], ||·||_{W^{1,2}}) → (C^0[0,1], ||·||_∞)$$ which maps every function to itself, and with Sobolev norm defined as $$||u||_{W^{1,2}}=||u||_{L^2}+||u'||_{L^2}.$$ ...
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$\frac{d}{dx}:C^1([0,1])\to C([0,1])$ as a closed, unbounded operator

First recall that a (potentially unbounded) operator $T:D(T)\subseteq X\to Y$ is closed whenever $(x_n)_{n=1}^\infty\subset D(T)$ convergent to $x\in X_0$ and $(Tx_n)_{n=1}^\infty$ convergent in $X_1$ ...
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Definition of ergodic map

I ask the similar question before. About definition of Ergodic theorem. Now just sincerely ask another fundamental problem about the definition of ergodic map. The following definition is what I ...
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1answer
34 views

Operator norm of Fourier transform operator

It may be trivial, but I am thinking the best way to show operator norm of Fourier transform operator on $L^1(\mathbb{R}^N)$ i.e. show $\Arrowvert T \Arrowvert =\frac{1}{{(2\pi)}^{N/2}}$. Since we ...
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1answer
62 views

$[T,S]:=TS-ST=I$ cannot holds

Let $\mathcal{B}(\mathcal{H})$ the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space $\mathcal{H}$. Let $T,S\in \mathcal{B}(\mathcal{H})$. I want to prove ...
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24 views

Strong continuity of $\langle Au,v \rangle=\int u^3 v dx$

I am currently trying to figure out the following. If I consider the space $W^{1,p}_0$ is it possible to show that the operator given by $$\langle Au,v \rangle=\int u^3 v dx$$ is strongly (weak to ...
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Adjoint of a polynomial in a closed linear operator.

Let $ H $ be a Hilbert space and let $ T $ be a closed densely defined linear operator in $ H $ with domain $ D(T) $ and with nonempty resolvent set. We define the following polynomial in T: $ P(T) :...
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1answer
54 views

$P,Q$ are Hermitian operators with only non-negative eigenvalues, then tr($PQ$) = 0 $\implies$ that $PQ = 0$.

So far I can prove: suppose there is a basis set $\{|i>\}$ that diagonalize $P$ with eigenvalue $p_i$, then tr($PQ$) = $\sum_i<i|PQ|i>$ = $\sum_ip_i<i|Q|i>$. Since $P, Q$ are ...
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Trace of a matrix exponential with tensor products, and Von Neumann entropy

$\def\T{\operatorname{Tr}}$ $\def\1{\mathbb{1}}$ Let $H=H_1\otimes H_2\otimes H_3$ be a finite dimensional Hilbert space, and let $\rho_{123}$ be a self-adjoint matrix with $\rho_{123}\geq 0$ (...
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1answer
30 views

Integral representation of log of operators

$\def\1{\mathbb{1}}$ Suppose we're in a "good enough" (finite for example) space, and we have positive (semi)-definite operators $P$ and $Q$. Let $\log{(P)}$ and $\log{(P)}$ be logarithms of $P$ and ...
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1answer
37 views

“bounding” an unbounded operator

I was wondering if, given a certain unbounded operator on a Hilbert space, it can (naively speaking) be "cutted" (or "bounded") by certain projections. So, thinking about this in a more sensible way, ...
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1answer
23 views

Understanding the defintion of dual operators

I'm reading a book about Functional Analysis, and now I've reached to the part about dual operators. I'm having some difficulties understanding the following definition - Why $A^*$ is $Y^*\...
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1answer
20 views

Triangle inequality for linear operators

For a linear operator $R \in B(X,Y)$ and some $k \in \mathbb{R}_{>0}$, do we have $$(\forall x \in X:\lVert R(x) \rVert \leq k \lVert x \rVert) \Rightarrow \lVert R \rVert \leq k $$? I'd like to ...
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1answer
21 views

Approximation a finite set with an operator

It is known that if $X$ is a compact manifold, then for every $\epsilon>0$, there is $\delta>0$ such that for every finite sequence $\{(x_i, y_i)\}_{i=0}^n\subset X$ with $x_i\neq y_i$ and $d(...
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2answers
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Compact operator $L:\ell^2\to\ell^2$ with $\Vert L\Vert=1$ such that $\Vert L(x)\Vert<\Vert x\Vert$ for all $x$

Let $\ell^2$ denote the space of square summable sequences of complex numbers. Let $L:\ell^2\to\ell^2$ be a linear operator with $\Vert L\Vert=1$ such that for all $x\in\ell^2\setminus\{0\}$, $\Vert L(...
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1answer
13 views

Supremum of norms of images of unit ball equals infimum of upper bounds of norms of images in the whole space

For a linear operator $T \in B(X,Y)$, there exists $k \in \mathbb{R}_{>0}$ such that for all $x$ in the unit ball of $X: \lVert T(x)\rVert \leq k $; $k \in \mathbb{R}_{>0}$ such that for all $...
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1answer
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Why Dirichlet form are interesting?

I'm currentely studing the Dirichlet form and to be honest, I really don't see in what they are useful. I don't really get the point with them. I recall the definition : Definition Let $(H,\left&...
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Sequence $(u_n)$ such that

Consider the following operator $H=\partial_{x^2}+x^2$ acting on $L^2(\Bbb{R})$. It's well known that the spectrum of $H$ is $\Bbb{R}$. So by weyl's criteron there exists $(u_n)\in D(H)$ such that $||...
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1answer
49 views

Corollary of the Hahn Banach theorem

I want to prove the following corollary of the Hahn-Banach theorem. Let $X$ be a normed space. For every closed linear subspace $Y\subseteq X$ and $x\in X-Y$, there exists $x'\in X $ such that $x'|Y=...
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1answer
60 views

Is a Rational Rotation Algebra a Cutdown of a Matrix Algebra?

Let $\theta=m/n$ and let $A_{\theta}$ be the rational rotation C$^{*}$-algebra with rotation angle $\theta$. I.e., $A_{\theta}=C^{*}(u,v)$, where $u$ and $v$ are unitaries such that $vu=e^{2\pi i \...
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prove/disprove if each two in $n$ operators can be diagonalizable simultaneously then all can be diagonalizable simultaneously

I have an idea that for $n$ diagonalizable operators $A_1, A_2, ..., A_n \in \ell(V)$. if each $A_i, A_j$ can be diagonalizable simultaneously then all of them can be diagonalizable simultaneously. ...
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1answer
23 views

Prove that there is no self-adjoint extension using deficiency indices

Consider an operator $P =-i\frac{d}{dx} : dom(P) \to L^2(\mathbb{R}^+)$ where $$ dom(P) = \{ f \in \mathcal{D}(\mathbb{R}^+) : f(0)=0\}$$ where $\mathcal{D}(\mathbb{R}^+)$ - smooth compactly ...
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1answer
34 views

Almost having invariant vectors vs having almost invariant vectors?

Let $\Gamma$ be a discrete and countable Group and let $\pi:\Gamma\to \mathcal{B(H)}$ be a unitary representation. We say that $\pi$ almost has invariant vectors if for every compact (=finite) subset ...
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2answers
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If $A = A^*$ then $ \lambda_{min} \leq\frac{\langle Av,v \rangle}{\langle v,v \rangle}\leq\lambda_{max} $

I'm leaning linear algebra and new to it. I have trouble with this problem and actually, I don't know what to do! any help or hint would be appreciated. for the linear operator $A\in \ell(V)$ wich $...
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1answer
53 views

if $AA^*=BB^*$ what are the relations between A and B [closed]

I'm wondering if we have two linear operators $A, B \in \ell(V)$. and we know that $AA^*=BB^*$. then what informations can this give to us about relationships between $A$ and $B$? I think they have ...
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1answer
51 views

If $T_n\to T$ strongly and $x_n\to x$ weakly, must $T_nx_n\to Tx$ weakly?

Let $X$ and $Y$ be Banach spaces, and let $\{T_n\}\subset L(X,Y)$, where $L(X,Y)$ denotes the space of bounded linear operators from X to Y.If $T_n\to T$ strongly and $x_n\to x$ weakly, must $T_nx_n\...
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1answer
40 views

Show that $(Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt$ is Compact linear operator on $C([0,1])$

Show that \begin{equation} (Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt \end{equation} is Compact linear operator on $C([0,1],R)$ where $\alpha, \beta:[0,1]\rightarrow [0,1]$ are continuous. My ...
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1answer
15 views

Stability of a linear equation

If $A$ is a matrix, then $e^{At} \leq C e^{-\lambda t}$ if and only if the spectrum of $A$ consists of eigenvalues with negative real parts. Is there a similar result, relating stability to the ...
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1answer
30 views

C*-algebra without finite-dimensional representations is simple?

Suppose $A$ is an infinite dimensional simple $C^*$-algebra. Then it has no non-zero finite dimensional representations. Is the converse also true? That is to say, if a $C^*$-algebra has no finite ...
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1answer
31 views

simple nuclear $C^*$ algebra [closed]

Does there exist an infinite dimensional simple nuclear $C^*$ algebra which admits a tracial state?
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41 views

Completely positive map is $*$-homomorphism

Suppose $A$ is a $C^*$ algebra and we have a completely positive contractive map $f \colon A\rightarrow B(H)$ such that $sup_{a,b \in A}\lVert f(ab)-f(a)f(b)\rVert =0$. Can we conclude that $f$ is a $*...
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2answers
48 views

What is the definition of a bounded operator in an infinite dimensional Hilbert Space?

I am struggling to understand the meaning of a bounded operator in a Hilbert Space. Does a bounded operator simply means that if it acts on an element of the Hilbert Space, the "result" is bounded?
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3answers
40 views

Is the Riemann integral a unary operator [closed]

Is the Riemann integral a unary operator and perhaps more importantly is this question even one that makes sense.
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1answer
36 views

strict topology on multiplier algebras

Suppose $A$ is a $C^*$ algebra,$M(A)$ is the multiplier algebra.If $S$ is a subset of $M(A)$ which is compact for the strict topology on $M(A)$,is $S$ also a subset of $M(M(A))$ which is compact for ...
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1answer
44 views

Operator norm of $T:l^{2}\rightarrow l^{1}$ where $Tx=(x_{1},x_{2}/2,x_{3}/3,x_{4}/4,…)$

As the title states, I need to compute the operator norm of a linear operator $T:l^{2}\rightarrow l^{1}$, where $$Tx=\left(x_{1},\frac{x_{2}}{2},\frac{x_{3}}{3},\frac{x_{4}}{4},... \right)$$ Using ...
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1answer
29 views

Compact operator on $L^2[0,1]^2$

Let $K\in L^2([0,1]\times[0,1])$, and we define the operator $T_k$ on $L^2[0,1]$. $$(T_kf)(x)=\int_{0}^{1}K(x,y).f(y).dy \quad \quad \forall f\in L^2[0,1]$$ How to prove that $T_k$ is a compact ...
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1answer
40 views

Spectral Theorem for Unitary Operator

It is well known that the following - in many literature - called the Spectral Theorem for Unitary Operator. I would like to know where i can find further information about it and its proof.
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1answer
24 views

Similarity of two self-adjoint operators

I am wondering whether the following is correct: Let $A$ and $B$ be two bounded self-adjoint, positive and invertible linear operators such that $\sigma(A)=\sigma(B)$ and $AB=BA$. Can we say that $A$...