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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Prove that the limit exist II

First question was here. I add one new condition. Let $T : H \rightarrow H$ is a linear continuous unitary ($T^*=T^{-1}$) operator, $H$ is a Hilbert space (not necessary). Suppose that $\forall h ...
3
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1answer
1k views

Prove that $T$ is an orthogonal projection

Let $T$ be a linear operator on a finite-dimensional inner product space $V$. Suppose that $T$ is a projection such that $\|T(x)\| \le \|x\|$ for $x \in V$. Prove that $T$ is an orthogonal projection. ...
3
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1answer
308 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 ...
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2answers
62 views

Verification of normality of linear operator

Let $X = (1,1,0),(1,-1,0),(0,0,1)$ be the basis of unitary space $\mathbb{C}^3$. Let $A$ be a linear operator, and $\mathbb{A}$ it's matrix in basis $X$: $$ \mathbb{A} = \begin{pmatrix}3 & i ...
4
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1answer
78 views

Prove that the limit exist

Let $T : H \rightarrow H$ is a linear continuous unitary ($T^*=T^{-1}$) operator, $H$ is a Hilbert space (not necessary). Suppose that $\forall h \in H \Rightarrow Th=h$ $T_n$ - a sequence of ...
3
votes
1answer
159 views

GNS-triplets for states on the matrix space: generalization to the infinite-dimensional setting

In a previous exercise, I have proven that states $\omega$ on the $C^*$-algebra $M_n(\mathbb C)$ correspond to a unique density matrix $\rho$ by the relation $\omega(A) = \mathrm{Tr}(\rho A)$. I was ...
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1answer
70 views

$‎‎\sigma(x)‎$ ‎‎‎‎is ‎contained ‎in ‎the ‎imaginary ‎axis ‎of ‎the ‎complex ‎plane

$‎A$ ‎is a‎ ‎C*-algebra ‎and ‎‎$‎x‎\in A‎$ ‎satisfies ‎‎$‎x‎^*=-x‎$.‎I want to show that ‎‎$‎‎\sigma(x)‎$ ‎‎‎‎is ‎contained ‎in ‎the ‎imaginary ‎axis ‎of ‎the ‎complex ‎plane.How i prove it?
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2answers
239 views

Projection operator for non-orthonormal basis

Let $V \subset H$ be Hilbert spaces. Let $\{v_j\}_{j=1}^\infty$ be a basis for $V$ and $H$. Define $V_N$ to be the span of $\{v_j\}_{j=1}^N$. We can define a projection operator $P:H \to V_N$ by ...
2
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1answer
47 views

limit of evaluated automorphisms in a Banach algebra

Let $\mathcal{A}=\operatorname{M}_k(\mathbb{R})$ be the Banach algebra of $k\times k$ real matrices and let $(U_n)_{n\in\mathbb{N}}\subset\operatorname{GL}_k(\mathbb{R})$ be a sequence of invertible ...
4
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213 views

Self-adjoint projections of a C*-algebra as complete lattices?

In Blackadar's Operator Algebras, there is the following remark after the proposition II.3.3.1 : The projections in a C*-algebra do not form a lattice in general In the answer of this question, ...
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45 views

Estimation of $\|(1-\Delta)^{\gamma/2}f\|_1$ type

Let $\phi(\xi)\in C_c^\infty(\mathbb{R}^d)$ with value $1$ in a neighborhood of the unit ball and vanishing fast outside the ball. I want to estimate $$\|(1-\Delta)^d (\cdot)^\alpha ...
3
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1answer
66 views

What is meant by $|dxdy|^{1/2}$ in the integral?

In this Daniel Grieser - Basics of the b-calculus paper the author mentions the term of a half-density on page 54 as an object which look like $u(x) |dx_1 \cdots dx_n|^{\frac{1}{2}}$. And I'm not ...
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1answer
68 views

An operator on $H\times H$, with $H$ Hilbert

Let $(H, \langle \cdot,\cdot\rangle_H)$ a Hilbert complex space and consider $H\times H$ with the inner product $$\langle (u,v),(z,w)\rangle_{H\times H}\ =\ \langle u,z\rangle_H + \langle ...
1
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1answer
228 views

Operator norm, convolution and Gauss-Weierstrass kernel

Let be $g_{t}(x)=\frac{1}{\left( 4\pi t\right) ^{\frac{1}{2}}}e^{-\frac{x^{2}% }{4t}},t>0,x\epsilon %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$ Gauss-Weierstrass kernel. For ...
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2answers
81 views

Relationship between two projectors

Please, somebody can help me with this problem? Let $V$ and $W$ be two closed subspaces of a Hilbert $(H, \langle \cdot,\cdot\rangle)$, and let $P:H\rightarrow V$ and $Q:H\rightarrow W$ the ...
3
votes
2answers
156 views

How to prove that operator is not compact in $L_2 (\mathbb{R})$

I have the operator $(Af)(x) = \int _{\mathbb{R}} e^{{-(x-t)^2}/2} f(t) dt$. It seems to me that it isn't compact and I'm looking for some general <=> criterion for integral operators to be ...
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1answer
50 views

Simple projector problem

Please, consider this ("sub")problem: Let $S$ a two-dimensional subspace of a Hilbert $H$ and let $Q\in\mathcal{L}(S,S)$, $Q\neq 0$ and $Q\neq I$, such that $Q^2 = Q$. Show that ...
4
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1answer
59 views

An operator between $\mathcal{L}(X, Y)$ and $\mathcal{L}(Y, X)$

Please, I need help with this problem. Let $X$, $Y$ be two vector normed spaces. Let $A_0\in\mathcal{L}(X, Y)$ such that $A^{-1}_0\in\mathcal{L}(Y,X)$. Show that there's an operator ...
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2answers
218 views

Closed operator

I've got a very straightforward question : if $T : B \rightarrow B$ is a linear continuous operator and $B$ is a Banach space, is $T$ a closed operator? This is obviously true in finite dimension, ...
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1answer
409 views

Matrix norm of a normal matrix

A normal matrix defined over a complex vector space has the property, that $\|A\|_2$ is its largest eigenvalue and now I was wondering whether this is also true for matrices defined over the real ...
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1answer
81 views

Norm operator and compactness

For the operator $U\colon \ell_{p}\to\ell_{p},\;\left( 1\leqslant p<\infty \right) :$ \begin{equation*} Ux=U\left( x_{1},x_{2},\dots \right) =\left( 0,x_{1},\frac{x_{2}}{2},\frac{% ...
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1answer
120 views

Closure of operator defined by extensions

I am study (first time) about closure of operator and when I was reading about it, I thought in question below. I don't know if this is true or false, I think is true, but I don't know how prove it or ...
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1answer
73 views

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
125 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
65 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
461 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 ...
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142 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
55 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. ...
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3answers
281 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
votes
2answers
189 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
156 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
69 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
80 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
244 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
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1answer
96 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
67 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
98 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
215 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
131 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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130 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
55 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
135 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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122 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 ...
0
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1answer
70 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 $\sigma(x^{-1})=\{\lambda^{-1} : \lambda\in \sigma(x)\}$
0
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1answer
48 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
124 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.