Questions tagged [compact-operators]

A compact operator is an operator from normed space $X$ to a normed space $Y$, such that image of every bounded subset of $X$ is relatively compact in $Y$. It's used with (functional-analysis) and (operator-theory) tags.

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Operator Compact

I would like a tip for this issue Let $K : [0, 1] × [0, 1] \rightarrow \mathbb{R}$ be a continuous mapping and $E = (C^{ 0}([0, 1]), ||.||_{\infty})$. Define $J : E \rightarrow E$ by $$Jf(x) = \int_{0}...
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Show that $f^{n}(x)\xrightarrow{n} x_0\forall x\in X$

$\text{Let $(X,d) $ be a compact metric space and $f:X\longrightarrow X$ a continuous operator}$ $\text{such that } d\big(f(x),f(y)\big)<d(x,y)\text{ } \forall x,y\in X, x\neq y.$ $\text{I know ...
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Strong operator limit of a sequence of compact linear operators is not compact

The statement of theorem: Let (Tₙ) be a sequence of compact linear operators from a normed space X into a Banach space Y. If (Tₙ) is uniformly operator converge say ∥Tₙ-T∥→0 then the limit operator T ...
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Is this operator T compact on $L^2([0,1])$?

Consider $L^2([0,1])$ and the operator $$Tu(t) = tu(t)$$for a.e. $t \in [0,1]$ How can I prove or disprove that T is compact? Thanks!
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Essential spectrum of operators whose resolvent difference is compact

Suppose that $T,S$ are densely defined, closed (unbounded) operators on a separable Hilbert space such that there exists $z \in \mathbb C$ in the intersection of resolvent sets of $T$ and $S$ for ...
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If $T$ is invertible and $K$ is compact, is $0\ne\lambda\mapsto(\lambda T+K)^{-1}$ holomorphic?

Let $X$ be a Banach space, $T\in\mathfrak L(X)$ be bijective, $K\in\mathfrak L(X)$ be compact and $$B:\mathbb C\to\mathfrak L(X)\;,\;\;\;\lambda\mapsto\lambda T+K.$$ Can we show that $B(\lambda)$ is ...
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What's the motivation for the term “compact” operator?

I am self-studying functional analysis and have just learned the definition of compact operators. However, it isn't clear to me why the name "compact operator" was chosen. The operator ...
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Div-Curl lemma and precompactness in $H^{-1}$

I am trying to understand $\operatorname{div-curl}$ lemma. An important requirement to apply $\operatorname{div-curl}$ lemma is the precomapctness of the sequences $\operatorname{div}(A_n)$ and $\...
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Spectrum of $C([0,2\pi])$ operator

Consider $T:C([0,2 \pi]) \rightarrow C([0,2 \pi]) $ $$ T(f(x)) =e^{ix} f(x). $$ Find spectrum, point spectrum and decide if $T$ is compact. To find point spectrum $\sigma_p(T)$ we need to find ...
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Show $T$ is compact operator if $\langle Te_n,e_n \rangle$ tend to zero.

Suppose $\mathcal{H}$ is a Hilbert space, and $T\in B(\mathcal{H})$. If for each orthonormal (norm 1) basis $\{e_n\}\subseteq \mathcal{H}$, we have $\langle Te_n, e_n \rangle \rightarrow 0$. Can we ...
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Are the functionals $\omega_{\xi_1, \xi_2}$ norm-dense in $B_0(H)?$

Let $H$ be a Hilbert space and for $\xi_1, \xi_2 \in H$ consider the bounded functional $$\omega_{\xi_1, \xi_2}: B_0(H) \to \mathbb{C}: x \mapsto \langle x \xi_1, \xi_2\rangle$$ Is it true that $$\...
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Is this operator on $L^2$ compact?

Let $T: L^2([0,1]) \rightarrow L^2([0,1])$ be a linear and bounded operator defined as: $$Tu(t) = \log(1+t)u(t) \space\space\space\space\space\space\space\space\space\space\space\space\space\space\...
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Is this operator a finite rank operator on $L^2$?

Is the operator $T: L^2([0,1]) \rightarrow L^2([0,1])$ defined as: $$Tu(t) = t u(t)$$ a finite rank operator? More in general: how can I understand if an operator $T$ on $L^2$ is a finite rank ...
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Exercise on bounded operator on L2 space

Let $H = L^2([0,3])$ and let $T \in B(H)$ defined by: $$Tu(t) = (1+t^2)u(t)$$ $||T||_{op} = 10$. i) Is T compact? Justify the answer. ii) Does $T$ have eigenvalues? Justify the answer. If you may ...
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Proof on sequence of finite-rank operators

I had difficulties figuring out the solutions to the following problem: Let $H$ be a separable infinite dimensional Hilbert space. Let {${L_n}$} be a sequence of linear finite rank operators from $H$ ...
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Justify that T is compact

I have troubles solving the following problem: Consider the Hilbert space $H=L_2([0,1],m)$ where m is the Lebesgue measure. Define $K:[0,1]\times [0,1]\to \mathbb{R}$ by: $K(s,t)=\begin{cases}(1-s)t\ \...
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No $T\in\mathcal{K}(X,Y)$ is onto [duplicate]

I have some struggles with this exercise. Let $X$ and $Y$ be infinite dimensional Banach spaces. Show that no $T\in \mathcal{K}(X,Y)$ is onto. We assume that $T\in \mathcal{K}(X,Y)$, so we have that $...
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Compact hermitian operator and eigenvalues

I would like to show that all non-zero eigenvalues of a (possibly unbounded) compact and hermitian operator $T:\mathcal{D}(T)\to \mathcal{H}$ on a seperable, infinite-dimensional Hilbert space have ...
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Idea of compact operators

I am studying functional analysis and in particular Hilbert spaces and operators, I would like some clarification on the idea of compact operators. The definition that was presented to me is: Given $\...
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Determining whether the set of eigenvectors form an orthonormal basis

I have a question concerning this question. In that case, can we say that the orthonormalized eigenvectors of the normal, compact operator T form an orthonormal basis for $H$? If not, under what ...
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If a linear operator maps to open unit ball to a relatively compact set, does it need to be bounded?

Let $X,Y$ be normed vector spaces and $A:X\to Y$ be linear and such that $AB^X_1(0)$ is relatively compact (where $B^X_1(0)$ denotes the open unit ball in $X$). Why can we conclude that $A$ is ...
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Spectrum and compactness of $L^1$ operator

Consider a continuous complex function $g$ defined on $[0,1]$. On $L^1([0,1])$ consider operator $$Tf(x) = g(x)f(x).$$ Calculate the norm. Determine its point spectrum and spectrum. Decide for what ...
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$T$ is compact iff every closed subspace in $Ran(T)$ is finite dimensional.

Let $T$ be a continuous linear operator on Hilbert space $\mathcal{H}$. Prove that $T$ is compact if and only if every closed subspace contained in the range of $T$ is finite dimensional. For the ...
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Is Biharmonic Operator m-accretive?

We have seen in "S. Zheng,Nonlinear Evolution Equations(Taylor & Francis, 2004)" that the laplace operator $\Delta$ is m-accretive. I wanted to ask whether biharmonic operator $\Delta^{2}...
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Compactness of $K(S)$, if $K$ is an infinite-dimensional compact operator in Hilbert space

$H$ is infinite-dimensional Hilbert space. $K: H \rightarrow H$ is infinite-dimensional compact operator. Let $S$ be a unit sphere in H. The task is to proof that $K(S)$ couldn't be compact. I know ...
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Compact operator, exercise 4.22 in Rudin's functional analysis

Suppose that $X$ is a Banach space and $T$ is a compact operator in $\mathcal{B}(X)$. $\lambda\neq 0$ and let $S=\lambda I-T$. Prove that: (a) $\mathcal{N}(S^n)=\mathcal{N}(S^{n+1}) \Rightarrow \...
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77 views

Compactness of $ Lu = -\partial_x^2 u + u $

I am trying to apply the spectral theorem to the operator: $$Lu = -\partial_x^2u + u $$ in $L^2([0,1])$ with domain $D(L)=\{u\in H^2([0,1]) s.t. \partial_xu(0)=\partial_xu(1)=0\}$. I need to prove ...
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66 views

Compact operator $T_{A,B}$

Let $A,B \in \mathcal{B}(X)$ where $X$ is a Banach space. Define $T_{A,B}: \mathcal{B}(X) \rightarrow \mathcal{B}(X)$ by $T_{A,B}(S)=ASB$. Prove that if $A$ and $B$ are compact then $T_{A,B}$ is ...
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75 views

Proving that $\cos(T) = \sum_{\lambda} \cos(\lambda) P_\lambda$ when $T$ is compact and self-adjoint

Let $H$ be a Hilbert space and let $T: H \to H$ be a compact and self-adjoint operator. Let $$T = \sum_{\lambda}\lambda P_\lambda $$ be its spectral decomposition (which converges in the operator norm)...
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l2 compact operator

Definined the mapping T: -> so that: where I'd like to show that if T is a compact operator then I thought i could negate my thesis and then show that T would not be compact but i wasn't able ...
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22 views

Existence minimal projections compact operators

Consider the following fragment from Davidson's book on $C^*$-algebras: Can someone explain why the sentence "Clearly, $P$ dominates a projection $E$ in $\mathfrak{A}$ with minimal positive rank....
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101 views

Weakly convergent nets and compact operators

Let $ K \in B(H)$ be a compact operator on a Hilbert space. We call a map $x: I \to H$ a net, if $I$ is a (not necessary countable) directed set. One can view nets as a generalization of sequences. ...
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Calculate Operator norm of a given operator

I have the following operator on a Hilbert space: $$T=\sum_{n=0}^{\infty}c_{n}\langle a_{n}\mid \cdot\rangle b_{n},$$ where the $c_{n}$ is a bounded sequence of complex numbers and $a_{n},b_{n}$ are ...
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Does left-multiplication by compact operators turn strong-convergence into norm-convergence?

If $\{T_i\}_{i\in I}$ is a bounded net of operators on a Hilbert space $\mathscr H$, converging strongly to some operator $T$, and if $K$ is a compact operator on $\mathscr H$, then the net $\{T_iK\}...
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51 views

Non-degenerate sub $C^*$-algebra of the compact operators.

Let $B_0(H)$ be the compact operators on the Hilbert space $H$ and let $B \subseteq B_0(H)$ a $C^*$-subalgebra that acts non-degenerately on $H$. Let $\{p_i: i \in I\}$ be a maximal family of pairwise ...
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Compact "Hilbert-Schmidt'' integral operators on Sobolev spaces

If one has an integral operator with kernel $K\in L^1(\mathbb{R}^2)\cap L^2(\mathbb{R}^2)$, then it is known as a Hilbert-Schmidt integral operator and it is compact on $L^2(\mathbb{R})$. My operator ...
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Compact operators in $\bigoplus_{i \in I} H_i$

Let $\{H_i: i \in I\}$ be a collection of Hilbert spaces. We can form the Hilbert space direct sum $$H:= \bigoplus_{i \in I} H_i$$ Question: Is there a "nice" dense subset of $B_0(H)$ (...
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Fredholm operators — index

Atkinson's theorem states: $T ∈ L(H)$ is a Fredholm operator if and only if T is invertible modulo compact perturbation, i.e. $TS = I + C_{1}$ and $ST = I + C_{2}$ for some bounded operator S and ...
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$B_0(H_1)B_0(H_2, H_1)$ is dense in $B_0(H_2,H_1)$

Let $H_1$ and $H_2$ be Hilbert spaces. Is it true that $$B_0(H_1)B_0(H_2, H_1)= \overline{\operatorname{span}}\{ST\mid S \in B_0(H_1), T \in B_0(H_2, H_1)\}= B_0(H_2,H_1)?$$ If so, how can I prove ...
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Compactness of diagonal operator $ T_\alpha(x_1, x_2,x_3…)=(\alpha_1x_1, \alpha_2x_2,\alpha_3 x_3,…)$

Let $\alpha=(\alpha_1, \alpha_2, \alpha_3...)$ a sequence of complex numbers that converges to zero. I need to show the compactness of the diagonal operator $T_\alpha:l^2\to l^2$ defined by $$ T_\...
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On stable isomorphism of separable nuclear C$^*$-algebras and Morita equivalence

I'm a new one to the theory of C$^*$-algebras, and I'm really missing something significant. According to Blackadar,"Operator algebras", page 153 Brown-Green-Rieffel theorem - For $\sigma$-...
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Norm of a certain operator in a Hilbert space

I am stuck on a particular problem. I consider the operator $$A:\mathbf{L}^2[-1,1]\longrightarrow\mathbf{L}^2[-1,1]:f\mapsto Af$$ where I define $Af(t) = t^2f(t)$, and I want to calculate its norm. I ...
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Does the sequence $v_{n+1} = T(v_n)/|T(v_n)|$ converge to an eigenvector?

Let $\varphi:[-1,1]\times \mathbb{R}\to\mathbb{R}$ the map $\varphi(x,w)=2\omega +1$ and $\mathcal{C}^0([-1,1])=\{f:[-1,1]\to\mathbb{R};\ f \ \text{is continuous}\}$ . I am interested in studying the ...
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Existence of Fredholm operators of a given index in a non-separable Hilbert space

Let $H$ be an infinite-dimensional Hilbert space. The (Fredholm) index of a bounded operator $T : H \to H$ is defined as $$ \mathrm{index}(T) = \dim \ker(T) - \dim \mathrm{coker}(T), $$ whenever the ...
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Show that two compact self-adjoint operators in Hilbert space must share at least one eigenvector [Stein Chapter 4 Exercise 35]

I am working on Stein Real Analysis, Chapter 4, Exercise 35 (a), which is an invariant of the spectral theorem. The exercise is stated as follows: If $T_{1}$ and $T_{2}$ are two linear self-adjoint ...
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25 views

Question about spectral theorem for compact operators

Consider the following fragment: Questions: (1) Does every non-zero eigenvalue occur in the $\{\lambda_i: i =1, 2, \dots\}$? (2) Why if $\lambda \ne 0$ is the eigenspace of $\lambda$ finite ...
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Compact operator with continuous inverse

Let $K:E \rightarrow E$ be a compact operator on a normed space $E$. Let $R^q$ be the range of $(\lambda I-K)^q$ where $q$ is such that $R^n = R^q$ for all $n \geq q$. Also, the restriction $(\lambda ...
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Compactness/non-compactness of a linear operator

I was fretting over this particular question which I found answered in the following link : https://math.stackexchange.com/a/1628972/817522 However now I have this notion that any sequence $T(x_n)$ ...
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37 views

Non-compactness of operator

Consider a linear operator $T$ on the Hilbert Space $l^2$ given by $$T(x_1,x_2,...)=(a_0x_1,a_1x_2,\cdots),$$ where the sequence $(a_n)$ is dense in $[0,1]$. I deduced that $T$ cannot be compact ...
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48 views

Bounded operators on complex Banach space $X$ are commutative exactly when $X$ is one-dimensional?

I am trying to prove that for a Banach space $X$ over $\mathbb{C}$, dim$(X)=1$ if and only if $\mathfrak{B}(X)$ is commutative. From this StackExchange question (Bounded linear operator commuting with ...

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