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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Stampacchia Problem

I need to solve this problem, but don't know how get that particular bound. Please, somebody can help me? Let $V$ a Hilbert space, $a : V\times V\rightarrow\mathbb{R}$ a bounded bilinear form, ...
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71 views

A theorem about operators in Hilbert sapce

For ench $n\geq1$, $B(\mathcal{H})$ is $\ast$-isomorphic to $\mathbb{M}_n(B(\mathcal{H}))$. Thanks to the one who tell me the proof or tell me where I can find the proof.
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1answer
30 views

$\{\hat f,\hat h\}=0$ and $\{\hat g,\hat h\}=0$ doesn't imply $\{\hat f,\hat g\}=0$?

I'm reading (Russian edition of) Landau & Lifshitz "Quantum Mechanics", and there is a statement like this (page 15 in link): We note that, if $\{\hat f,\hat h\}=0$ and $\{\hat g,\hat h\}=0$ , ...
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81 views

Generalized Gårding's inequality's problem

Please somebody can help me with this problem? I tried to solve it, but I couldn't do it, and I can not find nothing on the web. Let $H$, $Q$, $V$ and $W$ Hilbert spaces such that $H\subseteq V$ and ...
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1answer
116 views

On Fredholm operator

Consider operator $T: l^2(\mathbb{N})\to l^2(\mathbb{N})$ given by $T(x_1,x_2,\cdots)=(\lambda_1x_1,\lambda_2x_2,\cdots)$, where $\{\lambda_n\}_{n\in \mathbb{N}}$ is nonzero bounded complex numbers. I ...
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1answer
126 views

Lax-Milgram problem

I am trying to solve this problem: Let $H$ a Hilbert space, $A:H\times H\rightarrow\mathbb{R}$ a bilinear form, bounded and $H$-elliptic, and $F\in H^{\prime}$ ($H^{\prime}$ = dual space). Besides, ...
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34 views

What are the fixed point(s) of this mapping?

Given the mapping $\operatorname{A} \colon C[0;1] \rightarrow C[0;1]$ (functions which are continuous over a given interval) $\operatorname{A}x(t) = x^2(t)-x(t)-t^2;$ What are the fixed point(s)? ...
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157 views

How to find interesting operators for a quantum system?

How can we find "interesting" operators for a quantum mechanical system? I can think of the following method: Given some system with an associated Hilbert space $V$ and Hamiltonian $H:V\rightarrow ...
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1answer
93 views

What is an operator norm?

I came across this symbol where they are mentioning operator norms. I am not sure how it is different from the Frobenius norm. It was something like this: $|||\Omega-\hat{\Omega} |||_2$ where ...
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59 views

elliptic operator and wave front set

Let us $f(x) \in C^\infty $ on $\mathbb{R}^n$, and the pseudo-diff. operator $ Q$ is defined by: $(Qu)(x)=(2\pi)^{-n}\int_{\mathbb{R}^{n}}e^{ix\xi }f(x)\left | \xi \right |\hat{u}(\xi) d\xi$ Where ...
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75 views

Pseudo-differential operators

What is the meaning of the formula $\sigma (PQ)=\sum \frac{1}{\alpha!}\partial _{\xi }^{\alpha}pD_{x}^{\alpha}q\; ;\;\; \sigma (Q)=q,\;\;\; \sigma (P)=p$ if the series on right side is infinite? ...
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192 views

Prove a bilinear operator is symmetric and positive definite

I'm having problem showing the following: All operators are defined on $V$ which is real (not complex). Let $f$ be a bilinear operator that is anti-symetric (meaning $f(a,b)=-f(b,a))$ and let $J$ be ...
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29 views

an upper bound on Wave Front

Can you please help to understand how to solve this question: Let $f^{ij}(x)$ be a positive definite matrix smoothly varying with $x$ and define ...
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2answers
339 views

If $T$ is an operator on a complex inner-product space, each eigenvalue $|\lambda|=1$ and $||Tv||\le||v||$, show that $T$ is unitary.

If $T$ is an operator on a finite-dimensional complex inner product space, each eigenvalue $|\lambda|=1$ and $||Tv||\le||v||$, show that $T$ is unitary. Here's what I had in mind and where I was ...
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1answer
154 views

functional analysis in probability theory, Feller processes, contraction semigroups

In an entire year of probability theory coursework at the graduate level, there was only one time when functional analysis seriously appeared. That was ergodic theory. Now that my self-studies have ...
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293 views

Suppose $T^2$ is diagonalizable and $\ker{T}=\{0\}$, and every eigenvalue of $T^2$ is nonnegative. Show that $T$ is diagonalizable.

Suppose $T^2$ is diagonalizable and $\ker{T}=\{0\}$, and every eigenvalue of $T^2$ is nonnegative. Show that $T$ is diagonalizable. Of course $T$ is an operator on $V$. It seems to me that if I take ...
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1answer
60 views

Obtaining a bound on the differential operator

I just need a little bit of help filling in the missing details in the following passage from Reddy (1986)'s Applied Functional Analysis and Variational Methods in Engineering Let $C_0[0,1]$ be ...
3
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1answer
188 views

Does an irreducible operator generate an exact $C^{*}$-algebra?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : An operator $T \in B(H)$ is irreducible if $W^{*}(T)=B(H)$. Definition : A ...
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1answer
208 views

A question about Moyal product

In Geometric quantization theory the Moyal product is one of main tools. We know Moyal product for the smooth functions $f$ and $g$ on $ℝ^{2n}$ takes the form $f\star g = fg + \sum_{n=1}^{\infty} ...
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196 views

Does an irreducible operator generate a nuclear $C^{*}$-algebra?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : An operator $T \in B(H)$ is irreducible (Halmos 1968) if its commutant $\{ T\}'$ ...
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111 views

Resolvent is continuous

I try to show that the resolvent of an operator (unbounded) $T$ on Hilbert space is a continuous function of the complex variable. For that I derived the following: $R(z)-R(a)=-(z-a)R(z)R(a)$, where ...
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45 views

Why we need the measure of noncompactness

I just start studying the measure of noncompactness and I get confused. Why we try to measure the noncompactness of an operator? Is it to see if we can obtain a weak noncompactness?!
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61 views

Is every irreducible operator unitary equivalent to a banded operator?

This issue continues this question. Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an ...
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189 views

Sufficient conditions for a closed range of an injective operator extension

I am given Banach spaces $X$, $Y$, and a bounded linear operator $$ A\colon D(A) \subset X \to R(A) \subset Y, $$ with domain $D(A)$ dense in $X$ and range $R(A)$ dense in $Y$. Let the extension of ...
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1answer
185 views

Is every operator unitary equivalent to a banded operator?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis. $T \in B(H)$ is ...
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1answer
137 views

Bounded operator and dense sets

Let $A : E \rightarrow E$ be a linear operator, and $E$ a (typically infinite-dimensional) Banach space. Suppose that $S$ is a collection of norm 1 elements of $E$ spanning a dense subspace of $E$. ...
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114 views

Show that the operator $T$ is bounded and find the adjoint

I need help with this problem: Let $\{u_{n}\}$ and $\{v_{n}\}$, $(n \in \mathbb{N})$ be two different Hilbert bases in a Hilbert space $H$. Define a linear operator $T$ so that, for $x \in H$: ...
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1answer
115 views

Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space. Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} ...
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63 views

The classification of possible singular supports

I need to find the solutions of $D_{x_1}u=0$ on $\mathbb{R}^{n}$ and to classify the possible singular supports. Any one have an idea how to solve this kind of question? Thanks!
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140 views

Resource on Infinite Systems of Difference Equations

In my efforts (somewhere on the boundary of discrete mathematics and theoretical computer science), I have come upon the necessity of solving (or at least finding out some of the solution's ...
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1answer
152 views

Does the nontrivial commutants of the Volterra operator admit a strictly continuous spectrum?

Let $H$ be a separable infinite dimensional Hilbert space. Definition : The spectrum $\sigma(A)$ of $A \in B(H)$, is the set of all $\lambda \in \mathbb{C}$ such that $A - \lambda I$ is not ...
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1answer
206 views

Is there a nonnormal operator with spectrum strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space. Definition : An operator $A \in B(H)$ is normal if $AA^{*} = A^{*}A$. Definition : The spectrum $\sigma(A)$ of $A \in B(H)$, is the set ...
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64 views

Calculate the projection of $g(x)=\exp(−2x^2)$ onto the subspace $S$

I have problem to getting started on this one: "Let $f_1(x) = \exp(−x^2)$, $f_2(x) = xf_1(x)$, S the subspace of $L^2(\mathbb{R})$ spanned by $\{f_1,f_2\}$, and $P$ the projector onto $S$. Find $Pg$, ...
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71 views

Is there an operator such that the spectrum of all its nontrivial commutants is strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space. Definition : let $A \in B(H)$, then its spectrum $\sigma(A)$ is the set of all $\lambda \in \mathbb{C}$ such that $A - \lambda I$ is ...
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1answer
133 views

Is there an operator whose non-zero commutants are always injective?

Let $H$ be an infinite dimensional separable Hilbert space. Is there an operator $T \in B(H)$ such that, if $TA=AT$ with $0 \ne A \in B(H)$, then $A$ injective ? Bonus question : what is ...
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1answer
152 views

Norm of Hilbert's operator $H f(x)= \int_0^{\infty} \frac{f(y)}{x+y}\, dy$ [duplicate]

Hilbert's operator $$H f(x)= \int_0^{\infty} \frac{f(y)}{x+y}\, dy \quad\text{ for all } f \in {L}^2(0,+\infty) \text{ and } x \in(0,+\infty),$$ is regular integral operator on $L^2(0,+\infty)$ ...
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325 views

Is there an injective operator with a dense nonclosed range?

Let $H$ be an infinite dimensional separable Hilbert space. Is there an operators $A \in B(H)$ such that $Im(A) \subsetneq \overline{Im(A)} = H$ and $Ker(A) = \{0\}$ ? Bonus : We can build ...
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1answer
38 views

Extension of family of operators

Let $A(z)$ where $z\in \mathbb{R}$ be a family of (bounded) operators on some Hilbert space. Assume we know these operators have a meromorphic extension to all of $\mathbb{C}$. Assume moreover that we ...
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1answer
51 views

Restriction to $\mathbb{R}^{d-1}$ as an operator on $L^2(\mathbb{R}^d)$

Identify $\mathbb{R}^{d-1}$ with $\mathbb{R}^{d-1}\times \{0\}\subseteq \mathbb{R}^d$. Is there a bounded operator $T: L^2(\mathbb{R}^d)\rightarrow L^2(\mathbb{R}^{d-1})$ such that $T(\phi)=\phi ...
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327 views

extreme points of the unit ball of the Schatten classes?

Suppose $1<p<\infty$. What are the extreme points of the unit ball of the Schatten classes $S^p$? See below for the definition of $S^p$: http://en.wikipedia.org/wiki/Schatten_norm
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1answer
350 views

Norm of integral operator

Consider the operator $T(f(t)) = \int_0^t f(s)ds$, where $t \in [0,1]$, and $f(t) \in C[0,1]$. To prove $$\|T^n\| = \frac{1}{n!}$$ Thanks for suggestions.
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similarity between bundle shift

Let $E$ be a flat unitary bundle of rank $n$ over a domain $R$ in $\mathbb{C}$. It is known that bundle shift $T_{E}$ is similar to $T_{\mathbb{C^n}}$ (which is the bundle shift corresponding to the ...
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1answer
377 views

Matrix form of the differential operator $\sum_{k=1}^N x^k\frac{d^k}{dx^k}$

The following differential operator: $P(x,N)=\sum_{k=1}^N x^k\frac{d^k}{dx^k}$ is defined in $x\in\left[-1,+1\right]$. Is it possible to find a matrix form of this operator vs. $N$? Because it's ...
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1answer
142 views

Spectrum of linear operators

I can't solve the following: i) Let $T:l^2 \rightarrow l^2$ , $Tx=\{ (Tx)_n\}_{n=1}^{\infty}$ given by $$(Tx)_n = \dfrac{1}{2}x_{n-1} + \dfrac{1}{2}x_n.$$ Find $\sigma(T)$. ii) Let $S : l^2 ...
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1answer
292 views

Simple spectrum and the spectral theorem for bounded symmetric operators

I have a question regarding the spectral theorem for bounded self-adjoint operators. The book "Functional Analysis, an Introduction" by Eidelman, Milman, and Tsolomitis says that if an operator $T$ ...
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2answers
292 views

Strong convergence of operators

I'm working through the functional analysis book by Milman, Eidelman, and Tsolomitis, and I have a question. The book states a lemma that I'm a bit confused about: A sequence of operators $T_n\in ...
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1answer
625 views

Decomposition an operator in terms of symmetric and anti-symmetric components

In linear algebra, we can write any operator as the sum of a symmetric and skew-symmetric parts: $$A=A^{\mathrm{sym}}+A^{\mathrm{skew}}$$ where $$A^{\mathrm{sym}}=\frac{1}{2}(A-A^T)$$ and ...
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1answer
294 views

A theorem about operator theory

Define $$\operatorname{Ref}\mathcal{S}=\{T\in B(\mathcal{H}):Th\in[\mathcal{S}h], \forall h \in \mathcal{H}\},$$where $\mathcal{H}$ is a Hilbert space and $\mathcal{S}$ is a linear manifold of ...
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2answers
66 views

Self-adjoint extensions modern paper or book

Do you know some modern and recent paper, lecture notes, or book about self-adjoint extension theory (defect indeces, Von Neumann theory,...)? Classical references can also be helpful but I am ...
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1answer
141 views

An inequality involving operator and trace norms

Consider two square matrices $A, B \in \mathbb{R}^{n \times n}$ and let $\| \cdot\|_1$ and $\|\cdot\|$ be, respectively, the trace norm (the sum of singular values) and the usual operator norm (the ...