Questions tagged [operator-theory]

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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6answers
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Equivalent Definitions of the Operator Norm

How do you prove that these four definitions of the operator norm are equivalent? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for ...
15
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1answer
2k views

Show that $P_i$ and $\sum_i P_i$ being idempotent implies $P_i P_j=\delta_{ij}$

Let $X$ be a finite dimensional real linear space, or more generally a finite dimensional vector space over a field of characteristic $0$. Let $(P_i)_{i=1}^n$ be a finite sequence of linear mappings $...
17
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3answers
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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 $x,y\in\...
13
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2answers
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Exponential of a function times derivative

Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e. \begin{equation} e^{a\partial}f(x)=f(a+x) \end{equation} This can be easily verified from a Taylor series \begin{equation} ...
35
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4answers
31k views

Norm of a symmetric matrix equals spectral radius

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 ...
23
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3answers
9k views

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 ...
26
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1answer
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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?
14
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3answers
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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) = e^{a\...
8
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4answers
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Compact operators on an infinite dimensional Banach space cannot be surjective

I am reading a book about functional analysis and have a question: Let $X$ be a infinite-dimensional Banach-space and $A:X \rightarrow X$ a compact operator. How can one show that $A$ can not be ...
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2answers
503 views

Translating Tarski's Axiomatization/Logic of $\mathbb R$ to the Theory of Magnitudes

Update: This has become a project, but I need help. All answers will now be definitions, propositions, theorems, etc. that build on the theory. I will marks some of my own answers as community wiki so ...
3
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1answer
183 views

Is there a such thing as an operator of operators in mathematics?

Thus far I have seen operators of numbers and operators that perform on functions like Laplace, Fourier and Z-Transforms but is there an operator in existence that performs on other operators? Like a ...
7
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6answers
2k views

Bounded operator that does not attain its norm

What is a bounded operator on a Hilbert space that does not attain its norm? An example in $L^2$ or $l^2$ would be preferred. All of the simple examples I have looked at (the identity operator, the ...
6
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2answers
715 views

A characterization of trace class operators

Let $H$ be a separable Hilbert space and let $T\in B(H)$, such that $\displaystyle \sum_{j=1}^\infty\langle T\xi_j,\eta_j\rangle$ converges for any choice of orthonormal bases $\{\xi_j\}$, $\{\eta_j\}$...
24
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3answers
14k views

Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$?

I am a beginner of funcional analysis. I have a simple question when I study this subject. Let $L(X)$ denote the Banach algebra of all bounded linear operators on Banach space X, $T\in X$ is ...
18
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1answer
987 views

How is Laplace transform more efficient?

I wrote an answer on Laplace Transform, following a series of lectures by Prof.Ali Hajimiri (kindly take a look at the answer, my question is entirely based on that answer). In this answer, though I ...
28
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1answer
2k 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 ...
18
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2answers
6k views

Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$ A_f(g) = \int f(\tau)g(t-\tau)d\tau $$ and apply it to $g(t)=e^{ikt}$. ...
3
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1answer
3k views

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($A$) ...
22
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3answers
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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 \...
8
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1answer
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Norm of self-adjoint operator

I am trying to prove that $\|A\|=\sup_{||x||=1}|\langle x,Ax\rangle|$ for some selfadjoint bounded operator A on a Hilbertspace. Can anyone give me a hint how to prove it.
11
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2answers
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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: $$\...
8
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2answers
7k views

When is the image of a linear operator (between Banach spaces) closed?

Let $T: X \longrightarrow Y$ be a continuous linear map between two Banach spaces. When is $\operatorname{Ran}(T)$ a closed subspace? What theorems are there? Thanks :)
17
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2answers
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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 $\mathcal I$ ...
11
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1answer
2k 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.
5
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1answer
660 views

A linear map $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for some $T\in B(X,Y)$

As the title says, the question is how to prove Let $X,Y$ be Banach spaces. A linear map $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for some $T\in B(X,Y)$ The forward implication ...
5
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1answer
3k views

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 ...
34
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2answers
17k views

What is the difference between isometric and unitary operators on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is an identity operator, $^*$ is a binary operation.) What ...
20
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2answers
10k views

Spectrum of shift-operator

Hoi, consider the Hilbertspace $l^2$ and the Left and Right-shift operator \begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*} I know ...
21
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2answers
3k 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 ...
11
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2answers
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Resolvent: Definition

Given a Banach space $E$. Consider linear operators: $$T:E\supset\mathcal{D}(T)\to E:\quad T(\kappa x+\lambda y)=\kappa T(x)+\lambda T(y)$$ (No other assumptions on the operator!) Denote for ...
12
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2answers
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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 ...
9
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3answers
2k 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 Tx,y\...
11
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1answer
5k views

Proof Complex positive definite => self-adjoint

I am looking for a proof of the theorem that says: A is a complex positive definite endomorphism and therefore is A self-adjoint. Does anybody of you know how to do this?
8
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1answer
319 views

The ideal generated by a non-compact operator

I wanted to find a quick proof of the following well-known fact. Since I couldn't easily find a reference, I provide a proof below. Let $H$ be a separable Hilbert space, and $J\subset B(H)$ be a (...
5
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2answers
1k 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 an ...
5
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1answer
3k views

A strictly positive operator is invertible

Suppose that $H$ is an Hilbert space, and $T: H \to H$ is a self-adjoint strictly positive operator (i.e. $\langle Tx,x\rangle > 0$ for all $x \neq 0$). How do I show that this operator is ...
8
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2answers
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Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentially selfadjoint if it contains a dense subset of analytic vectors?
3
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1answer
950 views

Bounded operators on separable Hilbert spaces

Let $H$ be a separable Hilbert space. Show that every bounded operator from $H$ to itself can be approximated in the strong operator topology by a sequence of finite rank operators. Im not sure what ...
15
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1answer
837 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 ...
2
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2answers
847 views

Normal Operators: 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 ...
3
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1answer
294 views

How to prove Laplace transform is bounded on $L^2(\mathbb{R}_+)$?

The Laplace transform is defined by \begin{equation} (\mathscr{L}f)(s) \triangleq \int_0^\infty e^{-sx} f(x) dx, \quad s>0, \end{equation} then how can we check that the Laplace transform $\...
0
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1answer
153 views

Spectral Measures: Pushforward

This thread is Q&A. Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: $$...
10
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1answer
5k views

A linear operator between Banach spaces is weakly continuous iff norm continuous?

Claim : a linear function $T$ between Banach spaces is weakly continuous iff norm continuous? Okay, So I think I have realised weakly continuous implies norm continuous. As weakly continuous implies ...
12
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2answers
3k views

Hilbert space adjoint vs Banach space adjoint?

I have read that there are two 'options' for an adjoint when dealing with Hilbert spaces. Let $T : X \to Y$ be a bounded linear operator between the Hilbert spaces $X$ and $Y$. The Hilbert space ...
6
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1answer
4k views

Show $T$ is invertible if $T'$ is invertible where $T\in B(X)$, $T'\in B(X')$

Seems simple enough but I can't quite get it. $X$ is a complex Banach space, and $T\in B(X)$, $T'\in B(X')$ is its adjoint. Suppose $T'$ is invertible. How can we show that $T$ is invertible? I have ...
12
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1answer
2k views

What are the Eigenvectors of the curl operator?

The curl operator $\vec\nabla\times\mathbb{1}$ can be written as a skew-symmetric 3x3 matrix $$\mathrm{curl} = \begin{pmatrix}0 & -\partial_z & \partial_y \\ \partial_z & 0 & -\...
10
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1answer
2k 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, ...
10
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3answers
3k 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: $\rho(AB) \leq \rho(A)\rho(B)$ ?
10
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3answers
225 views

$\langle Tu\;|\;u\rangle=0,\;\forall u\in E \Longrightarrow T=0$?

Let $E$ be a complex Hilbert space. Let $T\in \mathcal{L}(E)$. I have two questions: Why it is not true that for an arbitrary operator $T\in \mathcal{L}(E)$, we have $\langle Tu\;|\;u\rangle=0,\;\...
9
votes
1answer
178 views

Absolut convergence implies being in trace class

I'm asking for your help in the next problem, I can't think how to do it! Prove that if $$\sum_{n=1}^{\infty}|(A\phi_n,\phi_n)|<\infty$$ for all orthonormal bases, then $A$ is in the trace class. ...

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