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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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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23 views

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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54 views

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 ...
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25 views

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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51 views

$T$ 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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35 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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52 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' ...
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1answer
60 views

Looking for a general approach to proving compactness of linear operators.

Preparing for my exam in functional analysis, I often have to prove that certain explicitly given operators are compact. I now have a decent amount of operators of which I can prove the compactness, ...
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44 views

How does the definition of compactness imply that all continuous operators are compact in finite dimensional spaces?

Let $S \subset X, Y$ be normed spaces over $K$. An operator $A:S \to Y$ is called compact if: $A$ is continuous $A$ transforms bounded set into relatively compact sets i.e. if $(c_n)$ ...
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36 views

Matrix representatin of a compact operator

Fact : I know for every compact operator $x : H \to H$, there are sequences $\{\xi_n\}$ and $\{\eta_n\}$ of orthonormal vectors of separable Hilbert space $H$, and sequence $\{\alpha_n\}$ in $\Bbb C$ ...
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21 views

Question about norm in trace class

I'm having troubles to proof the next inequality $$\|A\|\leq\|A\|_1$$ where A is an operator in the trace class and $\|A\|_1=Tr|A|$. And $\|A\|$ is the norm of operators. I just got this ...
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Proving that an operator $T$ on a Hilbert space is compact

Let $H$ be a Hilbert space, $T:H \to H$ be a bounded linear operator and $T^{*}$ be the Hilbert Adjoint operator of $T$. Show that $T$ is compact if and only if $T^{*}T$ is compact. My attempt: ...
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1answer
65 views

Existence of a surjective compact linear operator on an infinite dimensional Banach space

Does there exist a surjective compact operator $T:l^{\infty} \to l^{\infty}$ ? Even though this might be tagged as a repeat question, i still have some doubts that i would like to clarify. My ...
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35 views

Proving that an operator is Compact

I have to check that the following operator $T$ is compact: Define $T:l^{2} \to {l^2}$ by $Tx=y=(\eta_{j})$, where $x=(\zeta_{j})$ and $(\eta_{j})=\sum_{k=1}^\infty \alpha_{jk} \zeta_{j}$ and ...
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Compactness of a linear operator

The question is as follows: Show that a linear operator $T:X\to Y$ where $X$ and $Y$ are normed spaces is compact if and only if the image $T(M)$ of the unit ball $M\subset X$ is relatively compact ...
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44 views

Spectrum of compact operators on an infinite dimensional normed space

The question is as follows: Let $T:X \to X$ be a compact linear operator on a normed space. If the $dim X= \infty$ then show that $0 \in \sigma(T)$. My attempt: Suppose on the contrary that $0 ...
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How do you quickly show whether an operator is compact?

In my text on functional analysis, the author defines an operator on normed space as compact if it: continuous transforms bounded sets into relatively compact sets Okay, number 1 we can work with. ...
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55 views

Compact operators

From a textbook: Suppose $X$ and $Y$ are Banach spaces and $T : X \to Y$ is linear. Then $T$ is a compact operator if one of the following holds: (a) $\{Tx_n\}$ contains a convergent ...
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Compact diagonal operator

Suppose $A : H \to H$, where $H$ is a Hilbert space, is bounded. Also, $A$ is a diagonal operator with diagonal $\{a_n\}$. Show: If $A$ is compact, then $a_n \to 0$ as $n \to \infty$. Should I prove ...
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Is there an error in the solution for this exercise?

I have this exercise: H is a complex hilbert space. And T is a compact operator on H. Show that if H is not separable, then 0 is an eigenvalue of T. Hint: Use lemma 1, and theorem 2. The ...
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Can a positive definite kernel expanded as the product form with an arbitrary orthonormal system?

Notations mostly follow https://en.wikipedia.org/wiki/Mercer%27s_theorem. Mercer's theorem uses the eigenfunctions $\{e_j\}$ of the integral operator as the expansion function. I wonder if we could ...
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31 views

Weakly compact operator with different domains

Let $A$ be a Banach algebra. Suppose that $e\in A$ such that $e^2=e$ and $eAe$ is division algebra(i.e., $eAe$ is unital and every element of $eAe$ has inverse in $eAe$). Define $T_e:A\to A$ with ...
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Why's Daugavet equation important?

I've been recently studing Daugavet equation in $L^1[0,1]$ and $C[0,1]$. I understand most of the results I've found but I can't figure out why is it important to find operators that hold Daugavets ...
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Is $T$ a compact mapping from $W_{0}^{1,2}\left(\Omega\right)$ into itself? [closed]

Let $\Omega$ be an open bounded subset in $\mathbb{R}^{6}$ and $f$ be in $L^{8}\left(\Omega\right)$. For any $w$ in $W_{0}^{1,2}\left(\Omega\right)$, define $T\left(w\right)$ be in ...
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Show that $f \overset{T}{\rightarrow} \frac{1}{x} \int_0^x f(t) dt$ is Bounded, and is NOT Compact in $L^2(0, \infty)$

Can someone help me with this question? Let $f \in X = L^2(0, \infty)$, and define \begin{equation} (Tf)(x) = \frac{1}{x} \int_0^x f(t) dt \ . \end{equation} Show that $T$ is Bounded, and is NOT ...
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Show that $\{ x_n \} \overset{T}{\mapsto} \{ \sum_{k=1}^{\infty} a_{nk} x_k \}$ is compact

Can someone help me with this question? Let $\ell^2$ be the space of complex sequences $\{ x_1, x_2, \ldots \}$ that $\sum_{n=1}^{\infty} \lvert x_n \rvert ^2 < \infty$. If $\mu$ be Counting ...
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Adjoint of canonical expansion of compact operator

Lets say I have given a rank-$n$ operator $A = \sum^n_{k=1} \lambda_k \langle u_k, \cdot \rangle v_k$. Then it is straightforward to compute its adjoint as $A^\ast = \sum^n_{k=1} \lambda_k \langle ...
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Find eigenfunctions of the integral operator with kernel $\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny)$

Find the eigenvalue and eigenfunctions of the integral operator $Ku=\int_0^\pi k(x,y)u(y)dy$. $k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$. This is how I ...
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1answer
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functional analysis. Compact operator. Hilbert-Schmidt theorem.

I have the following problem: "Under which $ \alpha \in \mathbb{R}$ is the operator $ T: L_2 [1, + \infty) \to L_2 [1, + \infty) $: \begin{equation*} (Tf) (x) = x^{\alpha} \int_x^{\infty} ...
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104 views

Is there an example of a non compact operator whose square is compact?

Is there an example of a non compact linear operator T from a Banach space X to itself such that T^2 is compact? Of course the converse is true, as T ^2 is compact if T is. Here T^2 means T composite ...
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Powers of compact operators

Consider a Hilbert space $H$ and a compact self-adjoint operator $T : H \to H$. I want to prove that all positive powers (especially fractional powers) of $T$ are compact. From the spectral theorem, I ...
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Counter-examples of direct sum of compact operators on Banach spaces is compact

Given a Banach space $X$ which can be written a direct sum of two subspaces $Y\oplus Z$ and the $u\in B(X), w\in B(Y), v\in B(Z)$ and $u=w\oplus v$. where $B(\cdot)$ denotes the space of bounded ...
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Compact operators and functions

Assume we have a look to the space $L^2(S^1)$. An orthonormal basis of $L^2(S^1)$ is given by $p_n(x)=e^{2\pi in x}$ $(n\in\Bbb{Z})$. One can also have a look at the operator $S(p_n)=sgn(n)\cdot p_n$ ...
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Diagonal non-compact operator

Suppose we have an operator $I:l_2 \rightarrow l_2$ which is diagonal but not compact. Does that follow: there exists a constant $C$ such that infinite number of diagonal terms $>C$?
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Definition of a compact operator

Operator compactness is characterized by maps the send the unit ball to relatively compact sets. Does anyone have a good justification for why we call this property compactness? The best ...
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Convergence in strong operator topology and norm topology

Let $(T_n)\subset B(H)$ be a sequence of operators such that $T_n\to 0$ in strong operator topology. Show that $\|T_nK\|\to 0$ and $\|KT_n\|\to 0$ for every compact operator $K$. Let $f,g \neq 0 ...
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What is the intuition/motivation behind compact linear operator.

Compact Linear Operator is defined such that the operator will map any bounded set into a relatively compact set. Why is this property so special that it can be named as "compact"? Does it share some ...
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Prove an operator is compact and classify the spectral values

Let $T(\eta) = \{\frac{1}{2^n}\eta_n\}$ for $\eta = \{\eta_1, \eta_2, ...\} \in \ell_2$. I need to prove T is compact, and evaluate $\sigma_p(T), \sigma_c(T), \sigma_r(T)$. My feeling is that I need ...
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Product of compact, bounded and self adjoint operator.

$T \in B(H)$, and $T = S^2$ for some self adjoint operator $S \in B(H)$. I need to prove that T is compact if and only if S is compact. If S is compact, it is easy to show that T is compact since S ...
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Proof Check for Compactness of Integral Operator

Above is my question. I have completed the question, but I'm not 100% about my proof for the final part - it seems like I haven't done enough. I've shown that if $U$ and $V$ are compact, then so is ...
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Prove that an infinite matrix defines a compact operator on $l^2$.

Let $(a(i))_{i=1}^\infty$ be an absolutely summable sequence, i.e., $\sum_{i=1}^\infty |a(i)|<\infty$, and consider the infinite matrix $$A=\begin{bmatrix} a(1)&a(2)&a(3)&\cdots\\ ...
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Sufficient conditions on integration kernel for continuity of the integral operator

Suppose that we have a measure $d\mu(v)=e^{-|v|^2}dv$ on $\Bbb R^d$. We define a linear operator $$T[f](u)=\int_{\Bbb R^d} |u-v|^\beta f(v) d\mu(v).$$ I want to establish conditons on $\beta\in\Bbb ...
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On closed ranges and sequences which converge to zero

I'm reading a proof of the Fredholm alternative, and there is a claim that goes like this: Let $K:X\rightarrow X$ be a compact linear map. Define $T=I-K$, then $Y=\ker(T)$ is a finite dimensional ...
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135 views

Hilbert-Schmidt and compact operators

I am new to this site and i dont really know how to ask questions properly, so i am really sorry if i did something wrong. My question is if there is a way to prove that a Hilbert-Schmidt operator is ...
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How is the Point Spectrum of a Compact Operator Countable?

I'm working on understanding a proof that if an operator $A$ on a Hilbert space $\mathcal{H}$ is compact, then show that $\sigma(A) - \{0\} \subseteq \sigma_p(A)$. If you're not familiar with this ...