# 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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### Showing that a sequence of compact operators converge uniformly to their pointwise limit

I am working out of Phillipe G. Ciarlet's Linear and Nonlinear Functional Analysis with Applications and am struggling with exercise 4.9-5. The problem is as follows: Let $G$ be a function in the ...
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### Why is a compact linear operator also a bounded linear operator?

I know that a linear operator is bounded iff continuous, with these definitions of "bounded" and "continuous": a linear operator T (on H Hilbert space) is bounded if ∃M>0: ||Tf|...
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### dilatation analytic

I have seen that the spectrum of the operator $T=-\frac{d^2}{dx^2}+x^2$ is the $\{2n+1,n \in \Bbb{N}\}$ and by dilatation $x=r y$ the spectrum of the operator $T_r=-r^{-2}\frac{d^2}{dx^2}+r^2x^2$ is ...
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### Commutant of corner of $*-$algebra on a hilbert space is corner of commutant

Let $A$ be a $*-$algebra on a Hilbert space $H$ and $p$ be a projection in $A'$, where $A'$ is the commutant of $A$, that is, $$A':=\{u \in B(H): ua=au~\text{ for all }~a \in A\}.$$ If also $p \in A''$...
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### Proving that operator $x_n \to x_{n+1}/ n$ is compact

Prove that the operator $T:\ell^2\rightarrow\ell^2$ defined as \begin{equation} Tx = \bigg(x_2, \frac{x_3}{2}, \frac{x_4}{3}, \dots \bigg) \end{equation}for $x = (x_1, x_2, x_3, \dots) \in \ell^2$ is ...
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### operators that are compact and positive, their norm equal to the greatest eigenvalue.

I read this from a post Norm of integral operator in $L_2$ Can someone point me to formal proof of this statement? Thanks
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### Proving an operator is contractive

I am having trouble with the following problem: a) Let $H_n (x), n \in \mathbb{N}$ be Hermite polynomials associated to the measure $$d \mu = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}d \lambda$$ ...
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### Can the Volterra operator be norm-approximated by nilpotent operators?

Let $V$ be the Volterra operator on the Hilbert space $L_2[0,1]$. Then $V$ is bounded and quasinilpotent. Can $V$ be approximated in norm by nilpotent operators?
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### Spectrum of a compact operator - proof clarification

I'm reading through the book Linear Analysis by Bollobás. I'm having a bit of trouble understanding part of the proof of theorem 7 in chapter 13, which states: Let $T$ be a compact operator and ...
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### Compact integral operator on $H^1(\mathbb{R})$

Consider the operator $${\mathcal{L}}v=e^{-x}\int_{0}^x v(y)\, dy.$$ Is the operator ${\mathcal{L}}$ compact as an operator from $H^1({\mathbb{R}^+})$ to itself? To give some context to the problem ...
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I am studying for my exam in Functional Analysis but I'm confused about the following example: Consider $\mathcal{C}([-\pi,\pi],\mathbb{K})$ with the $\infty$-norm and define $T:X \to X$ by $$Tf(x)=\... 2 votes 1 answer 43 views ### Compactness of embeddings of Reproduing Kernel Hilbert Spaces with almost surely equal kernels Let \Omega \neq \emptyset and \mu be a probability measure on \Omega. Consider two reproducing kernels k_1,k_2:\Omega \times \Omega \rightarrow \mathbb{R}, such that they both represent the ... 2 votes 1 answer 43 views ### Prove composition of bounded and compact operators is compact. If T:X\to Y is a compact linear operator on Banach spaces X and Yand S:Y\to Z is a bounded linear operator where Z is a Banach space. Prove ST is compact. Here's my proof, let x_n \in X ... 0 votes 0 answers 30 views ### Decay rates of eigenvalues of Hilbert-Schmidt integral operator Let \Omega \subset \mathbb{R}^n be bounded. Suppose we have an integral kernel K: \Omega^2\to \Omega with \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|K(x,y)|^2dxdy < \infty. We know that the ... 0 votes 1 answer 59 views ### Prove that T is compact if and only if \mu_n \to \infty when n \to \infty Let \{e_n\} be an orthonormal basis for a Hilbert space H, T: H \rightarrow H be a bounded linear mapping and$$\mu_n = \sum_{x \perp \{e_1, \cdots, e_n\}, x \neq 0}\frac{\|T(x)\|}{\|x\|^2}$$... 0 votes 1 answer 41 views ### The equation x- T(x) = y has no solution for some y \in X. Construct a normed space X and a bounded linear operator T \colon X \rightarrow X with \|T\|<1 such that the equation x- T(x) = y has no solution for some y \in X. I know that if B is a ... 2 votes 1 answer 58 views ### Motivation behind the term, "precompact." Let X and Y be real Banach Spaces and let$$ K:X\rightarrow Y $$be a bounded linear operator. Consider a sequence \{u_k\}_{k=1}^\infty in X. We call a sequence \{Ku_k\}_{k=1}^\infty to be ... 2 votes 1 answer 27 views ### Why does E_2: = (I-T)E_1 with (I-T) being injective imply that E_2 \neq E_1? In chapter 6 of Brezis: Functional Analysis, in the proof on the "Fredholm Alternative," there is the following: T is a compact operator on E where E is infinite dimensional. Assume (... 0 votes 0 answers 36 views ### Trace of the projection operator Usually the trace of the orthogonal projection onto a finite dimensional space equals to the dimension of the space: Trace(P)=M where M denotes its dimension. For an orthonormal basis e_1,\cdots,... 0 votes 0 answers 35 views ### Is this operator compact in L^2[0,d]? Consider X=L^2[0,d] and define A \colon X \to X a linear bounded operator by the integral function \begin{equation} {A}x=\int_0^\cdot x(\xi)d\xi \end{equation} This operator looks compact to me ... 3 votes 2 answers 89 views ### Spectrum of sum of bilateral shift operator and a compact operator Let T be the bilateral shift operator, that is: T: \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z}) such that (T(x))_k=x_{k-1} (where x_{k-1} means the k-1 coordinate of the sequence. I have been ... 2 votes 1 answer 34 views ### Spectrum of sum of bounded and compact map Assume X is a Banach space and that T is a bounded linear map and K is a compact linear map from X to itself. I need to prove that that if \lambda is in the spectrum of T but is not an ... 0 votes 1 answer 38 views ### example in spectral theory Consider X=l_{2}. Let T : l_{2}\longrightarrow l_{2} be defined by : T(x_{1},x_{2},....)= (x_{1},\frac{x_{2}}{2},\frac{x_{3}}{3},...). And S=I , the identity operator. Here N(T)=N(S)=\lbrace{... 2 votes 1 answer 54 views ### Prove that a given operator is compact Let H be an Hilbert space with scalar product (\cdot,\cdot) and T : H \to H a linear operator defined as follow:$$ Tx = \sum _{n=1} ^\infty (x,a_n)b_n $$where (a_n)_{n\in\mathbb{N}},(b_n)_{n\... 1 vote 1 answer 67 views ### Essential Spectrum of arbitrary compact operator I need to find the essential spectrum of an abitrary compact operator. Let T : X \to Y be a compact operator. The essential spectrum is the set of all \lambda \in \mathbb{C} such that T- \lambda ... 1 vote 1 answer 41 views ### If \{\varphi_n\} is an eigenbasis of K^\ast K's support and h\perp K^m\varphi_n for all n,m\ge1, then how do you show Kh=0? \newcommand{\span}{\operatorname{span}}\newcommand{\im}{\operatorname{Im}}EDIT: According to the below answer, Royden's construction is wrong. However, the key detail which was omitted by the ... 0 votes 1 answer 19 views ### Zero is an eigenvalue of a compact operator on Non-Seperable spaces Let X be a non seperable Banach Space. If T is a compact operator on X, is it true that Ker(T)\neq \{0\}, i.e., 0 is an eigenvalue of T. This question was from my exercise, where X was ... 1 vote 0 answers 55 views ### The spectrum of T+C is stable under the perturbation of the form \frac An, where A and T are skew-adjoint and C is compact. Let \mathcal{H} be a Hilbert space. Let T and A be two skew-adjoint (unbounded) operators on \mathcal{H}, i.e., T^*=-T, A^*=-A. Also we have a bounded compact operator C on \mathcal{H}. ... 0 votes 0 answers 28 views ### Compact linear operator definition It is well known what we mean by a compact linear operator A:X\to Y where X,Y are Banach spaces (see https://en.wikipedia.org/wiki/Compact_operator#Compact_operator_on_Hilbert_spaces). I wonder ... 0 votes 1 answer 27 views ### How can I prove that the spectrum of the operator A\in B(C(K)) defined by Af = g∙f is equal to \text{Im}(g), for g\in C(K). I want to prove that \sigma(A), the spectrum of the linear operator A \in B(C(K)) which is defined by Af = g∙f, is equal to \text{Im}(g), for g\in C(K). We may assume that K is a compact ... 2 votes 0 answers 76 views ### How do I deal with the integration limits of the integrals that show up when showing thatKx(t)=\int_0^tK(t,\tau)x(\tau)d\tau, x\in X is compact? Let X=C(I) with the sup norm \|\cdot\|_\infty, where I=[0,1]. Let K be a Volterra integral operator:$$Kx(t)=\int_0^tK(t,\tau)x(\tau)d\tau, x\in X$$Show that K: X\rightarrow X, is compact ... 4 votes 1 answer 57 views ### Kernel of Bounded Operator on Hilbert Space I encountered this question on an exam recently and was not able to solve it. Suppose you have a bounded operator T on a Hilbert space \mathcal{H} such that I-T is compact, where I is the ... 0 votes 1 answer 34 views ### Equivalence of compact operators I'm reading about compact operators and I'm trying to prove the following statement: Let X,Y be Banach spaces and T:X \to Y a linear operator. Then the following are equivalent: (a) T is compact. (... 4 votes 1 answer 81 views ### Property of an operator that is Fredholm and compact? I have been asked this question at my course on Functional Analysis, to tell something about an operator that is both compact and Fredholm. The answer needs to be related to the spaces between which ... 2 votes 0 answers 53 views ### Infinite Matrices from l^p to l^{\frac{p}{p-1}} that are compact operators I wanted to ask if my proof (sketch) of the following statement is correct. Namely, let p>1 and define q= \frac{p}{p-1} we are given an operator K : l^{p} \rightarrow l^{q} defined as x \... 3 votes 1 answer 46 views ### Convergence in the operator norm Given the following operator: T:\ell^2\to\ell^2, which acts in the following way on the standard basis vectors:$$Te_{2k-1}=\frac{1}{k}(e_{2k-1}-ie_{2k})Te_{2k}=\frac{1}{k}(ie_{2k-1}+e_{2k})$$... 1 vote 2 answers 71 views ### Check whether operator T: L^1[0,1] \to L[0,1] is compact. [duplicate] I need to check whether the integral operator T:L^1[0,1] \to L^1[0,1] that sends f \to \int_0^x f(y)dy is compact or not. I am guessing it is not but I am struggling on proving it. My best shot I ... 1 vote 1 answer 43 views ### Nuclear Operator with Finite Dimensional Range I have a question regarding nuclear operators. Let me first give the definition: Let \Xi and H be real Hilbert spaces, and let T\in L(\Xi,H). T is a nuclear operator if there exists a sequence ... 0 votes 1 answer 40 views ### Prove that T is compact A \in B(X) and compact operator. Define a map T: X/\mbox{ker}(A) \to X. Where T defined as T(x + \mbox{ker}(A)) = A(x) for all x\in X. I want to prove that T is compact. I using the fact ... 0 votes 0 answers 18 views ### Compact identity operator is of finite rank It's well-known that the compact identity operator on Hilbert space is of finite rank. I'm wondering about the case on Hilbert A-module where A is a C^*-algebra. I feel it should be the same, ... 0 votes 1 answer 52 views ### Is either A or B a compact operator? X is a Banach space, consider two operator A,B\in B(X). Now product of two operator, i.e. AB is compact. Can we say that one of A and B is compact? I am just thinking in the following way; ... 3 votes 1 answer 146 views ### Check whether integral operator is compact Let T: X \to Y be the following integral operator, where f is just assumed to be integrable (not neccessarilly continuous):$$(Tf)(x)=\int_0^x f(t)dt In an exercise I have to check whether this ... 1 vote
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Let $X$ be Banach space, $N$ a (closed) subspace of $X$ and $\pi\in X\to X/N$ the natural quotient map. Call $Y\subseteq\mathbb{Z}\to X\sqcup X/N$ the set of functions on $\mathbb{Z}$ such that $f\in ... 1 vote 1 answer 37 views ### Question on pointwise and uniform boundedness. I'm reading through C* algebras by Murphy, and I'm covering compact operators at the moment, and the following was stated: "Let$X = C([0,1])$, and define$u \in B(X)$by$u(f)(s) = \int_{0}^1 k(...
I am studying functional analysis and got stuck with a textbook exercise. I greatly appreciate some hints/help or a push in the right direction! Define for $g\in C^0[-1,1]$ the integral operator \$T_g:... 