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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A simple adjoint operator question

I'm trying to solve this problem: Let $\Omega$ a bounded open of $\mathbb{R}$, consider the Hilbert real spaces $X = L^2(\Omega)$ and $Y = \mathbb{R}^{2\times 2}$, with the inner products: ...
4
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2answers
129 views

Prove that $A$ is bounded operator on $\ell^p$ and find $\| A\|$

For which one $p \ge 1$ is with $$A(x_n)_{n=1}^{\infty}=\left(\frac{1}{m}\sum_{n=1}^{m}\frac{x_n}{\sqrt{n}}\right)_{m=1}^{\infty}$$ defined bounded linear operator $A:\ell^p \to \ell^p$? Find norm ...
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0answers
67 views

Sum of Closed Operators

If $A$ and $B$ are two closed operators on a Hilbert space (not defined everywhere), is their sum closed as well? I think not, but cannot construct a counterexample. Some posts on this site do address ...
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1answer
492 views

Invertible operator

Let $K:V\to W$ such that $Kf = k$, where $V,W$ are infinite-dimensional Banach spaces. Is it correct to say that in general $f = (K^*K)^{-1}k$, however, when $V=W$, then $f = K^{-1}k$. $T^*$ here ...
3
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2answers
148 views

Representing the tensor product of two algebras as bounded operators on a Hilbert space.

Hi Math StackExchange, Let $A$ be a commutative, infinite dimensional, unital, *-algebra represented by bounded operators on a Hilbert space $H_A$. Next let $B$ be a finite non-commutative *-algebra ...
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2answers
56 views

Solution to operator equation, surjectivity

Suppose $T:V\to W$, where $V,W$ are banach spaces and $Tf = k$ (for instance $T$ might be an integral operator). They say that the equation has solution when $T$ is injective and so $T^{-1}$ exists. ...
6
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3answers
300 views

Prove $p^2=p$ and $qp=0$

I am not really aware what's going on in this question. I appreciate your help. Let $U$ be a vector space over a field $F$ and $p, q: U \rightarrow U$ linear maps. Assume $p+q = \text{id}_U$ and ...
4
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2answers
194 views

a trace class operator problem

Could someone help me with this Prove that If $A$ and $B$ are positive trace class operators on a Hilbert space, then so is $A^zB^{(1-z)}$ for a complex number $z$ such that $0 <Re(z)< 1$. An ...
2
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1answer
179 views

Normal operator matrix norm

I have some troubles to show that the operator norm of a normal operator is always equal to its largest eigenvalue, how can I proof this? Does anybody of you have a hint? My problem is, that I do not ...
6
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1answer
70 views

Is the adjoint operation WOT-WOT continuous?

This is a well-known property of the Hilbert-space adjoint operator that it is WOT continuous. Is a similar true for Banach spaces? That is, for a given Banach space $X$ is the operation ...
2
votes
1answer
81 views

Orthonormal functions as a combination of three complex functions

I have some problem to find three orthonormal functions in the interval $-1\le x\le 1$ as a linear combination of these three functions: $$f_1(x)=1,f_2(x)=x\exp(i\pi x),f_3(x)=\exp(i\pi x)$$ Is it ...
6
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2answers
268 views

Adjoint operator, bijective

Let $A\in\mathcal{L}(X,Y)$, where $X,Y$ are normed vector spaces. Define the adjoint operator $$\begin{array}{ll} A^{\prime}\ : & Y^{\prime}\rightarrow X^{\prime},\\ & G \mapsto ...
4
votes
1answer
109 views

Closure of operators

Let $X$ and $Y$ Banach. We say that the linear operator $A:\mathcal{D}(A)\subseteq X\rightarrow Y$ admits a closure if there's a linear operator $B:\mathcal{D}(B)\subseteq X\rightarrow Y$ such that ...
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1answer
60 views

Surjective function on product space

I know that, if $U$ and $V$ are closed subspaces of a Hilbert $(X,\langle\cdot,\cdot\rangle)$, then these statements are equivalents: $$i)\ U^{\perp}\subseteq U+V\quad\quad\quad ii)\ X = ...
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0answers
69 views

Is there a bijection there?

Let $X$ be a normed vector space and $T$ a subset of $X^{\prime} = \mathcal{L}(X,\mathbb{R})$. Then define the set: $$^{\circ}T\ :=\ \{\;x\in X\ :\ F(x)=0,\ \forall\ F\in T\;\}.$$ (When) Is possible ...
2
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2answers
104 views

Are there nonlinear operators that have the group property?

To be clear: What I am actually talking about is a nonlinear operator on a finitely generated vector space V with dimension $d(V)\;\in \mathbb{N}>1$. I can think of several nonlinear operators on ...
10
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1answer
224 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
133 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.
7
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1answer
141 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.
2
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2answers
85 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
47 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?
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0answers
60 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
136 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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131 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
73 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
50 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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3answers
104 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
130 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
135 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
40 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
90 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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1answer
210 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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90 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 ...
3
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1answer
199 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
287 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
62 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
149 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
78 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
126 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 ...
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1answer
40 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
184 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
337 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
107 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
165 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 ...
0
votes
2answers
179 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
135 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
161 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
134 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
votes
1answer
257 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
111 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 ...