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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Injectivity of index map for $K_1(S^1)$

This example/problem is from Valette's notes on the Baum-Connes conjecture (p. 45). The exercise is to prove that the (trivially equivariant) $K$-homology group $K_1(S^1)$ is $\mathbb{Z}$. For this, ...
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Problem understanding compact and Fredholm operators

I'm trying to understand the general interaction/duality between Fredholm and compact operators and I ran into the following: Let $L$ be the Laplacian or some elliptic operator on the Sobolev space $...
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If two compact, positive operators are close, are the projections onto subspaces also close?

Let $H$ be a Hilbert space. Let $a$ and $b$ be compact, positive operators acting on $H$. I wonder if the inequality $$\Vert \Pi_{\ker[a - \lambda_j(a)]}\, -\, \Pi_{\ker[b - \lambda_j(b)]}\Vert\leq \...
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Compact operators are orthogonally equivalent to a diagonal matrix?

On Brezis's Functional Analysis, the last question of Problem 44 (near the end of the book) reads (modified to include context) Assume that the Hilbert space $H$ is separable and $T\in\mathcal K(H)...
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$A$ and $A^*$ dissipative implies $D(A) \subset H$ is compact embedding

For selfstudy purpose I want to show the following: $H$ Hilbertspace, $D(A)$ dense subspace of $H$, $A\colon H \supset D(A) \to H$ linear closed dense defined operator. If $A$ and $A^*$ are both ...
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how is a compact embedding of infinite dimensional Banach spaces possible?

I'm looking at a dense defined closed operator $A\colon H \supset D(A) \to H$ with a Hilbertspace $H$ and $D(A)$ a dense subspace of $H$. In my notes there are some phrase like "if the embedding $D(A) ...
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Spectrum is finite or unbounded

Let $X$ be a Banach space and $A:D(A)\subset X\to X$ be a linear and closed operator with $\rho(A)\neq \emptyset$. Suppose that the map $$j:\left(D(A),\|\bullet\|_A\right)\hookrightarrow \left(X,\|\...
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Algebra of compact operators on $\ell_p$

Are the algebras of compact operators $K(\ell_p)$ and $K(\ell_q)$ isomorphic as Banach algebras for $1\leq p<q<\infty$?
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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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29 views

Multiplication functional $M_f$ compact iff $f$ is in $c_0(\mathbb{N})$

Let $f:\mathbb{N} \to \mathbb{C}$ be a bounded function and let $$ M_f : l^2(\mathbb{N}) \to l^2(\mathbb{N}) \hspace{0.2in} (M_fu)(n):=f(n)u(n) $$ be the correspoding multiplication operator. Show ...
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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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Finite rank operators are dense in compact operators on $L^p(\mu)$

Let $1\le p \le \infty$ and let $(X,\Omega, \mu)$ be a $\sigma$-finite measure space.If $A\in \mathscr{B_0}(L^P(\mu))$, show that there is a sequence $\{A_n\}$ of finite rank operators such that $||...
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50 views

necessary and sufficient conditions for certain operators on $C[0,1]$ to be compact

This is an excersice in Conway's 《A Course in Functional Analysis》. Let $\tau:[0,1]\rightarrow [0,1]$ be continuous and define $A:C[0,1]\rightarrow C[0,1]$ by $A(f)=f\circ \tau$. Give necessary and ...
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40 views

Compact operators form the only closed proper ideal of bounded linear operators

I am trying to understand the following proof in Trace Ideals and Their Applications by Barry Simon (Proposition 2.1): Let $\mathcal{J}$ be a two-sided ideal in $\mathcal{L}(\mathcal{H})$ containing ...
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38 views

Adjoint of Compact Operators in Normed spaces

I have a little trouble understanding adjoint operators in spaces without inner products. So the definition of the adjoint operator is the following: Definition (Adjoint operator $T^\times$): ...
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Compact operator in Hilbert spaces reach the maximum in the sphere.

I found the following question in my textbook: (QUESTION) Let $\mathcal{H}$ a Hilbert space and $T: \mathcal{H} \rightarrow \mathcal{H}$ a compact operator. Show that exists $x \neq 0$ in $\mathcal{H}...
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the c*-algebra generated by the Volterra operator

Let V be the Volterra operator on $\mathscr{L^2(0,1)}$.$V(f)(x)=\int_{0}^{x}{f(y)dy}$. Show that $C^*(V)$, the smallest C* algebra generated with V, is $\mathbb{C}+\mathscr{B_0(L^2(0,1))}$ where $\...
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34 views

Compact operator with no non-zero eigenvalues is zero?

Suppose we have a Hilbert space $H$ and a compact operator $T$ acting on $H$. If $T$ has no non-zero-eigenvalues, is it necessarily the zero operator? Secondly, if I decompose $H$ into eigenspaces of ...
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Spectrum of operator $T((x_n)_{n\in\mathbb{Z}})=\left(\frac{1}{n^2+1}(x_n-x_{-n})\right)_{n\in\mathbb{Z}}$

The eigenvalues should satisfy: $$T(x_n)=\lambda x_n$$ $$\frac{1}{n^2+1}(x_n-x_{-n})=\lambda x_n$$ $$\left[(n^2+1)\lambda+1\right]x_n=x_{-n}$$ I suppose that this should mean that $$\forall\lambda\in\...
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Trace-Operator is compact?

Assume $\Omega$ is a smooth, compact riemannian manifold with non-empty smooth boundary $\partial \Omega$. Let $T$ be the Trace-Operator $T \colon H^1(\Omega) \to L^2(\partial \Omega), \ f \mapsto f|_{...
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Bounds on eigenfunctions of integraloperator

Let $K: [0,1]\times[0,1] \to \mathbb{R}$ be a symmetric positive definit and continuous function. It is known my Mercer's theorem that $$ [T_K \varphi](x) =\int_0^1 K(x,s) \varphi(s)\, ds $$ is ...
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Dual operator of a compact operator

Why is the dual operator $A^{\ast}$ of a compact operator $A:X \rightarrow Y$, where $X,Y$ are two Banach spaces again compact?
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Different versions of Mercer's theorem

I am reviewing materials on reproducing kernel Hilbert space (RKHS) and I've found various versions of Mercer's theorem: About the positive-definiteness conditions. In the Wikipedia pages on RKHS ...
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Nontrivial closed ideal of $\mathbb{B(H)}$, $\mathbb{H}$ is a non-separable Hilbert space.

$\mathbb{H}$ is a non-separable Hilbert space. Give an example of nontrivial closed ideal $I$of $\mathbb{B(H)}$, that is different from $\mathbb{B_0(H)}$ which is the ideal of compact operators. Any ...
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Sequence of operators that commute imply the limit commutes?

Given a sequence of compact operators $A_n\to A$ as $n\ \to \infty$ and $B$ (which has finite rank). $\varphi \in L^2([a,b])$ If $A_nB\varphi = BA_n \varphi$ Am I able to say anything about $A$, i....
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48 views

$T:\ell^2 \to \ell^2$ is a compact operator

Any example in Kreyszig: Introductory Functional Analysis with Applications: Prove compactness of $T:\ell^2 \to \ell^2$ defined by $y=(\eta_j)=Tx$ where $\eta_k=\xi_j/j$ for $j=1,2,\dots$. Defined ...
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1answer
43 views

can a sum of two non-compact operators be compact?

I'm supposed to say, whether an operator $$Tf(t)=f(1-t)$$ can be expressed as $$T=\lambda I-K$$ where $K$ is a compact operator. As $T$ is not compact, I suppose that $K=\lambda I-T$ can't be compact ...
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Modified shift operator is compact.

For the operator $$T(\eta_j) = \frac{\eta_{j+1}}{j}$$ on Hilbert Space $H$ where $(\eta_j)$ is a basis. Show it is compact. Can this work? Define $$f = (\eta_j)_{j \geq 1}$$ $$T_N(f) = \left(\...
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Weak convergence = norm convergence for trace class operators?

Given a (separable) Hilbertspace $H$, I look at the traceclass operators $\mathfrak{S}_1$. I recall the fact that the weak convergence implies norm convergence in the sequence space $\mathcal{l}^1$. ...
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Compact operator for which the image of the closed unit ball is not compact [duplicate]

Do you have an example of a compact operator for which the image of the closed unit ball is not compact?
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29 views

compactness in $\ell^2$

How can I show T is compact when T is defined as $$ \text{T :}\,\ell^2 \to\ell^2\,\text{by Tx=y where} \,y_j=\alpha_jx_j\text{and}\,\alpha_j\to0\,\text{as}\,n\to\infty$$
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An operator satisfying in a sequence of equations

Assume that $H$ is a non-separable Hilbert space. Let $\{\eta_n\}$ be an arbitrary sequence in $H$. Let $\{\zeta_n\}$ be a sequence in $H$ which forms a linearly independent set. Does there exist ...
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bounded operator whose image is contained in the image of a compact operator

Let $A,K$ be 2 linear operators from $X$ to $Y$ (Banach spaces). $A$ is bounded and $K$ is compact. If $A(X)\subset K(X)$, is it true that $A$ is alse compact? I know that if the rank of $K$ is ...
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A translation invariant sigma algebra in $B(H)$

Assume that $H$ is a non-separable Hilbert space. Let $s_0$ be the family of all basic neighborhoods in the strong operator topology. We denote $M_s$ by the sigma algebra generated by $s_0$. ...
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Eigenvalues of an integral operator

The following operator is defined on $L_2(0,1)$: $$Kf(t)=\int_0^1|s-t|f(s)ds$$ I am wondering how I can calculate the eigenvalues and eigenfunctions of such an operator. I start with $\int_0^1|s-t|f(...
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28 views

Measure of non-compactness

Can someone give me some simple examples of measure of non-compactness of sets in Banach spaces or metric spaces, which are easy to understand.
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51 views

Compactness of a certain Integral Operator on $L^2$

Let $X\subset \Bbb R^n$ be a compact subset and $\alpha \in (0,n)$. Let $m:X\times X\to \Bbb R$ be a bounded measurable function. Consider the integral operator $$Tf(x)=\int_X \frac{m(x,t)}{|x-t|^\...
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Prove the following are equivalent.

Let $E$ and $F$ be Hilbert spaces. For, $T \in B(E,F)$, show that the following are equivalent: $(i)$ $T$ is compact $(ii)$ $T^*$ is compact $(iii)$ There exists a sequence of finite rank operators ...
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T compact if and only if T*T is compact.

I have an operator $T \in B(\mathcal{H})$. I need to prove that T is comapct if and only if $T^*T$ is compact. One way is ok, because if A or B is comapct then AB is compact, so I get at once that if ...
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Strong convergence due to Compact Operator [duplicate]

Given a sequence $u_{n}$ such that: $u_{n} \rightharpoonup 0$ in $L^{2}(\mathbb R^{n})$ & $A$ is a compact operator. The problem is to show that : $Au_{n} \rightarrow 0$ in $L^{2}(\mathbb R^{n})$ ...
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Are Hilbert-Schmidt operators in non-separable Hilbert spaces compact?

The definition of Hilbert-Schmidt operator should still be valid even when the Hilbert space is not separable: If $e_i$ for $i\in I$ is an orthonormal basis for a Hilbert space, and $\mbox{Trace}(T)=...
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Compact operator in terms of exact sequences

I know a pretty equivalent definition for Fredholm operators in terms of exact sequences. Here is it: We called operator $S : E \to F$ between Banach spaces $E$ and $F$ as Fredholm iff exist exact ...
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trace norm and tensor product

Let $(M_n (\mathbb{C}), n\|.\|)$ , $(M_n (\mathbb{C}), n\|.\|)$ and $(M_{nm} (\mathbb{C}), nm\|.\|)$ be three Banach algebras. where $$\|A\| = \mathrm{tr}\sqrt{(A^* A)}. $$ What is the norm of $\phi$ ...
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Functional Analysis, infimum of $A+K$ where $K\in K(X)$

This is a problem from Martin Schechter's Book (Principles of Functional Analysis) If $X$ is a Banach Spaces and $A\in B(X)$, let $$|A|_K=\inf_{K\in K(X)}\|A+K\|.$$ Show that $|A|_K<1$ implies ...
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$T$ is a compact operator but not a finite rank operator

Prove that if $X$ is a Banach Space and if $T$ is a compact operator but is not a finite rank operator, then $0 \in \bar {T(S_X)}$(where $S_X$ is the unite sphere). Please provide some solution (...
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36 views

Prove compactness of an operator.

Suppose $$ X=\left\{x \in C^2(\Bbb R,\Bbb R):x(t+T)=x(t)\; \text{for all}\;t \in \Bbb R \right\}, $$ $$ Y=\left \{h \in C(\Bbb R,\Bbb R):h(t+T)=h(t)\;\text{for all}\;t \in \Bbb R \right \} ,$$ and ...
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56 views

C*-algebraic intrinsic definition for compactness of an operator?

Some properties of operators (normal, self adjoint, hermitian) have intrinsic definitions for any element of a $C^*$-algebra. Is there such definition for compact operators? Equivalently: Let $\...
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$\mathcal{A}+K$ is norm-closed where $\mathcal{A}$ is a $C^*$-algebra and $K$ is the compact operators.

Let $\mathcal{A}\subset B(H)$ be a unital $C^*$-algebra and let $K$ be the closed ideal of compact operators. I need to show that $\mathcal{A}+K$ is also a $C^*$-subalgebra of $B(H)$. I am stuck at ...
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Compactness of Single Layer operator

The single layer operator defined by $S: H^{-1/2}(\Gamma) \rightarrow H^{1/2}(\Gamma), \, Sf(x) = \int_{\Gamma} G(x,x') f(x') dx'$ with $G(x,x')$ the Green's function and $H^s$ the Sobolev space of ...
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Compactness of the trace operator

Is it true that for a set $\Omega$ with Lipschitz boundary the trace operator $T : H^1(\Omega) \to L^2(\partial \Omega)$ is compact? Can you please give a reference? I found a theorem in Necas' ...