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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273 views

Criteria of compactness of an operator

Suppose $K$ is a linear operator in a separable Hilbert space $H$ such that for any Hilbert basis $\{e_i\}$ of $H$ we have $\lim_{i,j \to \infty} (Ke_i,e_j) = 0$. Is it true that $K$ is compact? ...
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1answer
143 views

Hilbert space proof

$X$ is a separable Hilbert space and $ A\in L(X,X)$ and compact. I need to prove that $A$ is approximately of finite dimension.
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178 views

Trace of a differential operator

Given the differential operator: $$A=\exp(-\beta H)$$ where $$H=\frac{1}{2}\left( -\frac{d^2}{dx^2}+x^2 \right)$$ and $\beta\gt 0$ How can I get the trace of this operator? Thanks in advance.
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2answers
86 views

Continuous Linear Mapping $C[0,1]\rightarrow C[0,1]$

Show that $L(f)(x)= \int_0^x f(t) dt $ is a continuous linear mapping from $C[0,1]$ into itself. Do I only have to show that the operator is bounded? How to do I explicitly choose my $M$ such that ...
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1answer
48 views

$(P\Lambda P^{-1}=T^2)~\implies~(\exists \Lambda'~\text{s.t.}~T=R\Lambda' R^{-1})$: $\;P,R\;$ Unitary Matrices

Let $T$ be a linear operator such that the operator $T^2$ is diagonalizable. Is $T$ necessarily diagonalizable?
2
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0answers
64 views

How to compress a linear operator and have the lossless composition property.

Consider a linear operator on $\mathbf{R}^n$ represented by a square matrix of size $n \times n$, call it $A$. The matrix acts on a row vector, call it $x$ and returns a row vector, call it $x'$, so ...
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1answer
145 views

Let $A$, $B$ be positive operators in a Hilbert space and $\langle Ax,x \rangle=\langle Bx,x \rangle$ for all $x$, show that $A=B$

Let $A$ and $B$ be positive operators in a Hilbert space $H$, and suppose that $\langle Ax,x\rangle=\langle Bx,x\rangle$ for every $x$ in $H$. Show that $A=B$.
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152 views

If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism?

$X$ and $Y$ denote Hilbert spaces. If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism? Homeomorphism means continuous map with continuous inverse. I think the ...
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1answer
85 views

The spectrum of an invertible element $x$ is $\sigma(x^{-1})=\{\lambda^{-1}: \lambda\in \sigma(x)\}$

Suppose $x$ is invertible in the unital Banach algebra $A$. How can I prove that $\sigma(x^{-1})=\{\lambda^{-1} : \lambda\in \sigma(x)\}$
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1answer
52 views

Spectrum of T in $B(\ell^2)$

Let $T:\ell^2 \to \ell^2$ be an operator on $\ell^2$ is defined as follows: $$T\{a_1,a_2,\dots\}=\{0,a_1,a_2,\dots\}$$ What is spectrum of T in $B(\ell^2)$?
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109 views

Diagonalizable Operators: An Operational Extension

Let $T$ be a diagonalizable operator on a vector space $V$. Prove that the operator $$a_nT^n + a_{n-1}T^{n-1}+\cdots+a_1T+a_0 Id_V$$ on $V$ is also diagonalizable for any scalars $a_1, ...
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1answer
186 views

Selfadjoint operator $\Rightarrow$ Idempotent Operator?

If $P\in\mathcal{L}(H,H)$, with $H$ a Hilbert space, such that $P = P^*$, Is possible to show that $P^2 = P$? If that is possible, then $P$ is a projection operator, right? Thanks in advance.
4
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1answer
155 views

Adjoint operator, a condition for closed range

Let $X$ and $Y$ be two Hilbert spaces and $A\in\mathcal{L}(X,Y)$. Suppose that there's $\beta > 0$ such that $$\inf_{z\ \in\ \text{Ker}(A)}\|x-z\|\ \leq\ \beta\|A(x)\|,\quad \forall\ x\in X.$$ Show ...
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0answers
42 views

question about idempotent operators [duplicate]

Let $H$ be a Hilbert space and $\dim(H)=\infty$. If each $T\in B(H)$ is finite sum of idempotent operators? If each $T\in B(H)$ is infinite sum of idempotent operators?
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1answer
121 views

Composition of Partial Isometry

Let $T$ be a linear operator in $H$, a Hilbert space. An operator $T \in L(H)$ is said to be a partial isometry if the restriction of $T$ to $ker(T)^{\perp}$ is an isometry. I would like to prove that ...
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262 views

Generalized eigenspaces of a compact operator are finite dimensional

Let $T : H\rightarrow H$ be a compact operator on a Hilbert space $H$. Say that $\lambda \in \mathbb C$ is a generalized eigenvalue of $T$ if there is some $n \geq 1$ such that $(\lambda - T)^n$ is ...
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104 views

Concerning unbounded linear operators on a Hilbert space

Let $H$ be some Hilbert space and let $B:H\rightarrow H$ be a bounded linear operator and $T:H\rightarrow H$ an unbounded linear operator. Furthermore we assume that $T$ is closed ,i.e. it's graph in ...
4
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1answer
230 views

Conditions for a kernel of a bounded operator to be complemented

I am well aware of the problem of complementing subspaces in Banach spaces as it was discussed here and here . Nevertheless, I wonder whether there are conditions for existence of a complement $M$ ...
3
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2answers
302 views

What is my operator norm (cannot get good enough bounds).

Given a space of square integrable functions $x(t)$ over the interval $[0;1]$ one can introduce a norm $$\|x(t)\|= \sqrt{\int_0^1 (x(t))^2 \, dt};$$ Then what is a norm of the transformation below ...
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1answer
74 views

Density of the image and closedness of the inverse of a bounded linear operator

Let $A \colon X \to X$ be a bounded linear operator, where $X$ is a Banach space. $(Q1)$ Is it true that if $A$ is injective then the image of $A$ is dense in $X$? $(Q2)$ Is it true that $A^{-1} ...
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0answers
173 views

Bounded operator on dense subspaces

Give an operator like this or show it doesn't exist: Operator $T: X\rightarrow Y$ is bijective. $X,Y$ are dense subspaces of a Banach space $Z$, and $X$ is proper subset of $Y$. Both $T$ and $T^{-1}$ ...
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1answer
85 views

Composition of $\mathrm H^p$ function with Möbius transform

Let $f:\mathbb D\rightarrow \mathbb C$ be a function in $\mathrm{H}^p$, i.e. $$\exists M>0,\text{ such that }\int_0^{2\pi}|f(re^{it})|^pdt\leq M<\infty,\forall r\in[o,1)$$ Consider a Möbius ...
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1answer
147 views

Bounded and invertible operator on dense subspace

Who can give me an operator like this or show it doesn't exist: Operator T: X-->Y, is a bijection from normed linear space X to normed linear space Y. X, Y are equipped with the same norm, and X is a ...
2
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1answer
42 views

Range of adoint operator

We consider infinite dimension. $X,Y$: Banach Spaces $T:X→Y$ is a bounded linear operator. I want to prove $(\ker\, T)^\bot = \overline {R(T^*)}$. $(\ker\, T)^\bot = \{f\in X^*|f(x)=0\ (x\in ...
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1answer
231 views

What is the adjoint of $x + \frac{d}{dx}?$

I have solved other problems like this using integration by parts. In this case, I can't figure out what to make each part for the integration. The question is true/false. Ultimately to show this you ...
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2answers
453 views

Coercive bilinear form on Hilbert space

I need to show the two following results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance. Consider a continuous symmetric bilinear form $B$ on a ...
3
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1answer
121 views

$AB - BA = I$ in Hilbert Space [duplicate]

Let H be a Hilbert space and $A$ and $B$ be bounded operators in $H$. How can I prove that $AB - BA = I$ is not possible ? Probably this is as easy as in the matrices case, but I couldn't prove it. ...
4
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1answer
205 views

A problem on bounded invertible linear operator in Banach space

Let $X$ be a Banach space. Let $T : X \to X$ be a invertible linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all $k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x ...
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225 views

Self adjoint operator

I am looking in the space of test functions $ \{f \in C^\infty|f^{(n)}(a)=f^{(n)}(b)=0\};n \in \mathbb{N}_0\} $whether the n-th derivative is a self adjoint operator. the dot product is given by ...
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1answer
158 views

Factoring a time derivative operator outside of an integral in space

I'm trying to integrate $$\int_a^b \frac{d}{dt} \left[ \frac{du}{dx}\right]dx.$$ Assume $u$ is a sufficiently smooth function of both $t$ and $x$. Since the integral operator is in space only, can ...
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2answers
212 views

Representation of a bilinear form on an Hilbert space

Given a bilinear symmetric form $b(u,v)$ on a Hilbert space. I need to know some very basic facts. A reference where these are discussed would be greatly appreciated. 1) There exists a symmetric ...
2
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1answer
144 views

Computing an explicit solution to an integral equation via the Neumann Series.

I am hoping that someone can provide guidance for solving the integral equation $$u=f+\lambda Au$$ where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
2
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1answer
328 views

Spectrum of the unbounded operator $i\partial_x$

I've been puzzling over this for some time now, and can't quite make my intuitions precise. I need to find the resolvent set and spectrum of the operator $$ Lu=i\frac{du}{dx} $$ taken to be ...
2
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1answer
127 views

K-theory, $K_{0}$ of algebra of compact operators

I don't understand how to define the trace of a matrix with values in operators. This occurred in the following situation: Suppose that $H$ is an Hilbert space and $K$ is the algebra of compact ...
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1answer
236 views

Residual spectrum is empty

I'm following Kreyszig's "Introductory Functional Analysis with Applications" and am trying to follow the proof of the following Theorem (9.2-4 on p. 468) For a bounded self-adjoint linear operator ...
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0answers
91 views

Find spectrum of the operator and an explicit form for the solution

Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if $v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$ (a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum (b) ...
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0answers
163 views

Find spectrum of the operator

I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem. Let $\theta \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator $Au=v,$ ...
3
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1answer
193 views

Convergence of operator norm

I have a linear bounded operator $A:L_2(0,1) \rightarrow L_2(0,1)$ satisfying $\|A^n\|^{1/n} \rightarrow 0$. Thus, for some sufficiently large $N$, $\|A^N\| < 1$ and then from Gelfand's formula, I ...
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1answer
393 views

Is my proof of characterisation of self-adjoint operators on complex Hilbert spaces okay?

I wish to show the following theorem: Let $T:H\to H$ be a bounded linear operator on a complex Hilbert space $H$. Then if $\left\langle Tx,x\right\rangle \in\mathbb{R}$ for all $x\in H$, then $T$ is ...
3
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1answer
183 views

Extension of a Bounded Operator on $L^p$ to $L^r$

Let $1<p\leq \infty$, and let $p^{-1} + q^{-1} = 1$. Let $T$ be a bounded operator on $L^p$ such that $\int(Tf)g = \int f(Tg)$ for all $f,g \in L^p \cap L^q.$ Show that $T$ uniquely extends to a ...
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Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
3
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1answer
203 views

The set of compact linear operators is a subspace of the set of bounded linear operators

I know that a linear operator $T:X \to Y$ (where $X$ and $Y$ are normed vector spaces) is compact if for every sequence $\left(x_{n}\right)\subseteq X$ s.t. $\left\Vert x_{n}\right\Vert \leq C$, the ...
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2answers
94 views

Sequence of operators in a Hilbert space

The question is: Let $H$ be a Hilbert space and $\{T_n\}$ be a sequence in $B(H)$ such that $\lim_{n\rightarrow\infty}\langle x, T_n y \rangle = 0$ for all $x, y \in H$. Prove or disprove $\sup_n ...
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1answer
64 views

Showing a bound on a contour integral

I'm working through M. Schechter's 'Principles of Functional Analysis' and I'm working through a proof on page 136 that shows that the spectral radius $r_{\sigma} (T) $ of a bounded linear operator ...
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1answer
259 views

When are two commuting linear operators functions of each other

I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up. If we formally consider the integral operators ...
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2answers
84 views

Limit of bounded operators

Suppose $T_n$ is a sequence of self-adjoint bounded operators on a Hilbert space, and $T_n \rightarrow T$ in operator norm, $T$ being also bounded and self-adjoint. Do we then have: $T_n^m\rightarrow ...
2
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0answers
55 views

Find a symbol for pseudodifferential operator

$\DeclareMathOperator{\Mel}{M} \newcommand{\Rn}{\mathbb R^n} \newcommand{\dd}{\,\mathrm{d}}$ Consider a pseudodifferential operator (Mellin operator) in positive orthant with symbol $\sigma(z)$: $$ ...
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1answer
67 views

Orthogonality & Adjoint Operator

I am trying to prove this simple statement left to the reader in Brézis's book. Let $A \colon D(A)\subset E \longrightarrow F$ be an unbounded operator. Let $G:=\operatorname{Graph}(A)$ and $L=E ...
7
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1answer
299 views

Approximating a Hilbert-Schmidt operator

Let $H$ be a separable Hilbert space. Recall that a bounded operator $A : H \to H$ is said to be Hilbert-Schmidt if $$\|A\|_{HS}^2 := \sum_{i=1}^\infty \|A e_i\|^2 < \infty$$ where ...
12
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3answers
271 views

Is there an algebra of summable series?

Let $D$ denote a divergent series and let $C$ denote a convergent series. Furthermore, let $s : \{ Series \} \to \{ numbers \} $ be a regular, linear divergent series operator, which is either one of ...