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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227 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 $M$ ...
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2answers
298 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
72 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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171 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
82 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
141 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
223 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
423 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
115 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. ...
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196 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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2answers
209 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
154 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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207 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
142 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
313 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
126 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
214 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
88 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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158 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 $Au=v,$ ...
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1answer
186 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
378 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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180 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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428 views

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
185 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
93 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
63 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
249 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
83 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 ...
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54 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
290 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 ...
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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 ...
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1answer
97 views

product of bounded linear operators

If I have 2 bounded linear operators $T_1,T_2$ such that $T_2:X\rightarrow Y$ and $T_1:Y\rightarrow Z$. I know that by boundedness, $||T_2(x)||\leq||T_2||\,||x||$ and using the norm of $T$ defined as ...
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1answer
89 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 ...
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437 views

What is the role of supremum in operator norm

An operator norm is defined as $\|A\|_S=\sup\{\|Av\|:v\in \Bbb R^n, \|v\|=1\}$. Where $\|\cdot\|$ is some norm on $\Bbb R^n$ and $A\in M_n(\Bbb F)$, space of square matrices of dimension $n$ over ...
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1answer
55 views

Prove that the only operator on $\mathbb{C}$ for which his inner product is zero is zero

How do I prove this statement: Let $V$ be an unitary vector space over $\Bbb C$, $(,)$ be an inner product on $V$ and $\Bbb A$ operator $V\rightarrow V$. Then $(\Bbb Av,v)=0$ if and only if $\Bbb ...
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1answer
109 views

spectrum of two bounded linear operators

Suppose that L and B are bounded linear operator on H, assume $0\in \rho(L) \cap \rho(L+B)$ and that $L^{-1}$ is compact. Prove that L+B also has a compact inverse.
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1answer
177 views

Bounded linear operator in weak topology

Let $B$ be a bounded linear operator on $H$. Prove $B\colon (H,w)\to (H,w)$ is continuous. $(H,w)$ is a Hilbert space with its weak topology.
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1answer
106 views

Is the Strong Limit of a Linear Operator in a Hilbert Space the Same as the Norm Limit?

If $H$ is a Hilbert Space, and I have an operator $F:H \rightarrow H$ which is the limit of a sequence of operators $F_n$ with respect to the operator norm; and this same sequence of operators ...
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1answer
352 views

Is the sum of two normal operators normal?

A normal operator is defined as $AA^{*}\ =A^{*}A$ Where A is an operator how do i show the sum of two normal operators is normal? Or find a counter example that shows this is false?
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80 views

Linear functional $\mathscr{L}(E,F)$

Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$. Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question: How to prove ...
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1answer
149 views

Proof Involving Difference Operators

Let E be the forward shift operator on $x$ defined by $Ef(x) = f(x+1)$. Similarly, let $\delta$ be the forward difference operator such that $\delta f(x) = f(x+1) - f(x)$ and the inverse operator ...
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2answers
224 views

Interchanging closed operators and integrals

I am dealing with a problem in Evans PDE without measure theory knowledge... We have contraction semigroup $\{S_t\}_{t \geq 0}$ on real Banach space $X$, i.e family of bounded linear operators from $ ...
7
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1answer
79 views

How does $\lim A_n$ being not invertible imply $\sup_n\|A_n^{-1}\|=\infty$?

Consider a sequence of operators $\{A_n\}_{n=1}^{\infty}\subset B(X,Y)$, where $X,Y$ are normed vector spaces and $B(X,Y)$ denotes the space of bounded linear operators from $X$ to $Y$. Assume that ...
5
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1answer
193 views

Matrix Representation of Trace Class Operators

Suppose we have a separable Hilbert space (thus with a countable basis) and that represent an operator in matrix form, i.e: $A: H \rightarrow H $$$x \;\rightarrow \sum_{j \in \mathbb{N}}\left(\sum_{k ...
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1answer
230 views

Relation between noncommutative geometry and functional analysis

Recently I came across the subject of noncommutative geometry via my interest in functional analysis. My very little exposure to this subject gives me a sense that part of it is built on the theory of ...
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166 views

How to decompose a representation into direct sum of cyclic representation?

Let $U$ be the bilateral shift operator on $l^2 (\mathbb Z)$, let $T=U+U^*$. How to calculate $\sigma(T)$? And how to show there is no cyclic vector for the action of $C^*(T,I)$. Further how to ...
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140 views

How can projection operators be limits of powers of unitary operators?

Consider a (fixed) unitary operator $U$ acting on the Hilbert space $\mathcal{H}$. Because the unit ball is compact in the weak topology, it is not hard to see that there exists a (smallest) compact ...
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1answer
872 views

Find norm of the integral operator

Find norm of the following bounded linear operator $$Ax(t)=\int_0^1e^{-ts}x(s)ds$$ where $x\in C[0,1]$ and $t\in[0,1]$. Please help me.