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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How to prove that an operator is compact?

Consider $T\colon\ell^2\to\ell^2$ an operator such that $Te_k=\lambda_k e_k$ with $\lambda_k\to 0$ as $k \to \infty$ how to prove that it is compact?
4
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Question about Fredholm operator

$X,Y$ are Banach spaces and $A\in B(X,Y)$ is a Fredholm operator (that is, the dimensions of ker($A$) and coker($A$) are both finite), then are closed linear subspaces ker($A$) and Im($A$) ...
20
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919 views

Why do zeta regularization and path integrals agree on functional determinants?

When looking up the functional determinant on Wikipedia, a reader is treated to two possible definitions of the functional determinant, and their agreement is trivial in finite dimensions. The first ...
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Compactness of a bounded operator $T\colon c_0 \to \ell^1$

Pitt Theorem says that any bounded linear operator $T\colon \ell^r \to \ell^p$, $1 \leq p < r < \infty$, or $T\colon c_0 \to \ell^p$ is compact. I know how to prove this in case $\ell^r \to ...
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Easy Proof Adjoint(Compact)=Compact

I am looking for an easy proof that the adjoint of a compact operator on a Hilbert space is again compact. This makes the big characterization theorem for compact operators (i.e. compact iff image of ...
6
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1answer
476 views

Sum of Closed Operators Closable?

Let $A$ and $B$ be closed operators on a (separable complex) Hilbert space with dense domains $D(A)$ and $D(B)$ respecitvely. Then, we may define the operator $A+B$ on $D(A)\cap D(B)$. In general, ...
9
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367 views

Question about Angle-Preserving Operators

This an exercise out of Spivak's "Calculus on Manifolds". Edit: There was a typo in the exercise as is noted below in the answers. The statement has been edited to reflect this. Given ...
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925 views

Compact sets as point spectrum of a bounded operator

It is well known that if $K$ is any compact set in $\mathbb{C}$, then there exist a bounded linear operator $T:l_2\to l_2$ such that $\sigma(T)=K$. My questions are: Q1) Does there exist $T$, a ...
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Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?

Using the Taylor expansion $$f(x+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dx^k}f(x)$$ one can formally express the sum as the linear operator $e^{a\frac{d}{dx}}$ to obtain $$f(x+a) = ...
12
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871 views

Closure of the invertible operators on a Banach space

Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ ...
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Equivalent Definitions of the Operator Norm

Would you give me a proof of the equivalence of these ones? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ ...
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What is operator calculus?

I watched the excellent interview with Richard Feynman: http://www.youtube.com/watch?v=PsgBtOVzHKI In the interview Feynman mention that he at young age re-invented operator calculus. I have searched ...
3
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278 views

Resolvent: Definition

Given a Banach space. Consider linear operators: $$T:\mathcal{D}(T)\to E:\quad T(\kappa x+\lambda y)=\kappa T(x)+\lambda T(y)$$ (No other assumptions on the operator!) Denote for shorthand: ...
3
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1answer
275 views

Graph of symmetric linear map is closed

A homework problem: Let $H$ be a Hilbert space. Let $T:H\rightarrow H$ be a symmetric linear map ($\langle Tx,y\rangle=\langle x,Ty\rangle$). Show that $S$ is bounded. My attempt: I'd ...
6
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1answer
520 views

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $. Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
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A compact operator is completely continuous.

I have a question. If $X$ and $Y$ are Banach spaces, we have to prove that a compact linear operator is completely continuous. A mapping $T \colon X \to Y$ is called completely continuous, if it maps ...
0
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2answers
122 views

Normal Operators: Spectrum vs. Numerical Range

Disclaimer: As I realized in the comments that this works for normal operators I decided to modify this question. Besides, I got the proof now - thanks to T.A.E.! Prove that for normal operators the ...
8
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4answers
348 views

Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question... Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty ...
3
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2answers
928 views

$\operatorname{Range}T$ is a closed subspace.

Let $X,Y$ two Banach spaces. If $T \in \mathcal{B}(X,Y)$ study if $\operatorname{Range}T$ is a closed subspaces. How can I prove this fact ? What theorems can I use ? thanks :)
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If $(I-T)^{-1}$ exists, can it always be written in a series representation?

If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm < $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$. Thinking in terms of a ...
7
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$T$ surjective iff $T^*$ injective in infinite-dimensional Hilbert space?

Let $T:H_1\rightarrow H_2$ be a bounded linear operator where $H_1$ and $H_2$ are Hilbert spaces. The Hilbert-adjoint is defined to the the operator $T^*:H_2\rightarrow H_1$ such that $\langle ...
4
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1answer
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Compactness of unit ball in WOT of B(X)

It is known that the unit ball in $\mathcal{B}(H)$, where $H$ is a separable Hilbert space is compact in the weak operator topology. Is it the same true if instead of $H$ we have any separable Banach ...
3
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Hilbert's Inequality

Could you help me to show the following: The operator $$ T(f)(x) = \int _0^\infty \frac{f(y)}{x+y}dy $$ satisfies $$\Vert T(f)\Vert_p \leq C_p \Vert f\Vert_p $$ for $1 <p< \infty$ where ...
3
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Hellinger-Toeplitz theorem use principle of uniform boundedness

Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all ...
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Some question about extension of bounded linear operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$H: ...
0
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1answer
109 views

Spectral Measures: Lebesgue

Preface Dominated convergence: $$f_n(\omega)\to f(\omega)\quad(\omega\in\Omega)\implies f_n(E)\to f(E)$$ (This gives a tool for analysis of operators.) Problem Given a Borel space $\Omega$ and a ...
17
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3answers
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Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
11
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2answers
1k views

Gelfand-Naimark Theorem

The Gelfand–Naimark Theorem states that an arbitrary C*-algebra $ A $ is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. There is another version, which states that ...
8
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648 views

Spectral radius inequality

Suppose $A,B \in M(n \times n, \mathbb{C})$ or $ A,B \in M(n \times n, \mathbb{R}) $. Under wich hypothesis can I state that: $\rho(AB) \leq \rho(A)\rho(B)$ ?
6
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local convexity of $L_p$ spaces

wiki says The spaces $L_p([0, 1])$ for $0 < p < 1$ are equipped with the F-norm they are not locally convex, since the only convex neighborhood of zero is the whole space Why is this so? ...
3
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1answer
118 views

Positive operator is bounded

For a real Banach space $X$ let $A:X\rightarrow X^*$ be a positive operator in the sense that $(Ax)(x)\geq 0$ for all $x\in X$. Show that $A$ is bounded. I don't know how to do that, maybe it's ...
6
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2answers
271 views

$\exp(A+B)$ and Baker-Campbell-Hausdorff

A few years ago, I did research in quantum mechanics, specifically dealing with generalized displacement operators. In such musings, BCH lights (or gets in, depending on your viewpoint) the way. A ...
4
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1answer
199 views

Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$

Let $A$ be a bounded or unbounded selfadjoint linear operator on a complex Hilbert space $H$ with spectral representation $A=\int_{\sigma}\lambda \, dE(\lambda)$ given by the Spectral Theorem for ...
4
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1answer
649 views

positive invertible operators

I need the following result. I think it's quite obvious but I don't know how to prove that: Let $C, T : \mathcal{H} \rightarrow \mathcal{H}$ be two positive, bounded, self-adjoint, invertible ...
3
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2answers
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Norm of a matrix equals greatest eigenvalue

How do I prove that the norm of a matrix equals the absolutely largest eigenvalue of the matrix? This is the precise question: Let $A$ be a symmetric $n \times n$ matrix. Consider $A$ as an ...
3
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1answer
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Prove that $T$ is an orthogonal projection

Let $T$ be a linear operator on a finite-dimensional inner product space $V$. Suppose that $T$ is a projection such that $\|T(x)\| \le \|x\|$ for $x \in V$. Prove that $T$ is an orthogonal projection. ...
3
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1answer
878 views

Compact multiplication operators

In class, we started talking about operators on Banach spaces after covering the Arzela-Ascoli Theorem. We defined a continuous operator $T\colon X \to Y$ to be compact if $\overline{T(B_X)}^{Y}$ is ...
2
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1answer
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Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
11
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1answer
375 views

Every Hilbert space operator is a combination of projections

I am reading a paper on Hilbert space operators, in which the authors used a surprising result Every $X\in\mathcal{B}(\mathcal{H})$ is a finite linear combination of orthogonal projections. The ...
8
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414 views

Proof that operator is compact

Prove that the operator $T:\ell^1\rightarrow\ell^1$ which maps $x=(x_1,x_2,\dots)$ to $\left(x_1,\frac{x_2}{2},\frac{x_3}{3},\dots\right)$ is compact. For an arbitrary sequence $x^{(N)}\in\ell^1$ ...
7
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3answers
228 views

If a linear operator has an adjoint operator, it is bounded

This is a question I'm struggling with for a while: Let $H$ be a Hilber space. Let $T,S: H\rightarrow H$ be linear operators (not neccessarily bounded) such that for every $x,y\in H$: $\langle ...
7
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3answers
821 views

Spectral measure of the multiplication operator

I have the following question: let $(X,\mathcal B,\mu)$ be a finite measure space and consider the operator $T\colon L^2(X,\mu)\to L^2(X,\mu)$ given by $Tf(x)=\varphi(x)f(x)$, where $\varphi\colon ...
6
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1answer
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Weak* operator topology and finite rank operators

We will say that ${T_i}\subset B(X,Y^*)$ converges to $T$ in W*-operator topology if $T_i(x)\rightarrow T(x)$ in W*-topology of $Y^*$( $\forall y\in Y \langle T_i(x),y\rangle \rightarrow \langle ...
6
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1answer
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Questions about a PDE: $-u_{txx}+u_{t}=u_{x}$, $u(x,0)=u_0(x)$, $x,t\in{\bf R}$

Consider the BBM equation: $-u_{txx}+u_{t}=u_{x}$, $u(x,0)=u_0(x)$, $x,t\in{\bf R}$. One may rewrite this equation as following $u_t=((I-A)^{-1}\partial_x)u$ where $Au=u_{xx}$ if $(I-A)^{-1}$ ...
4
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1answer
272 views

Norms involving positive operators

Let's say we have $A \leq B$. Is it then true that $||Ax|| \leq ||Bx||$ (where $x, A, B$ all belong to the same finite-dimensional Hilbert space $H$)?
3
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Alternative definition of strong/weak operator topology.

Given two normed spaces $(X, ||\cdot||_X)$ and $(Y,||\cdot||_Y)$ the space of bounded linear maps $\mathcal{B}(X,Y)$ can be equipped with the strong operator topology (SOT) as follows: The ...
3
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2answers
935 views

Spectral radius of the Volterra operator

The Volterra operator acting on $L^2[0,1]$ is defined by $$A(f)(x)=\int_0^x f(t) dt$$ How can I calculate the spectral radius of $A$ using the spectral radius formula for bounded linear operators: ...
3
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67 views

Projection of the third dual of a Banach space onto the first dual

Let $j_X:X\rightarrow X^{**}$ denote the canonical embedding. I've read several articles where it is assumed that the reader is familiar with the idea that there is a norm one projection from ...
3
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1answer
508 views

Operators on $C([0,1])$ that is compact or not.

For $f\in C([0,1])$ set $$Hf(x) = \frac{1}{x}\int_0^x f(t)dt.$$ a) Show that $H$ is a bounded operator from $C([0,1])$ into itself which is not compact. b) From a) it follows that $H$ induces a ...
3
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1answer
450 views

Weakly compact operators on $\ell_1$

Is the following assertion true/known? Let $V$ be a Banach space and let $T\colon \ell_1\to V$ be a bounded linear operator. Is it true that $T$ is not weakly compact if and only if there is a ...