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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What is a biorthogonal system?

What does biorthogonal mean ? If they say let the system $l^1,l^2,l^3,...,l^n$ Be biorthogonal to the bases $x_1,x_2,x_3,...,x_n$ Of the kernel of $Λ$ So that $Λ$ Is a linear operator
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Finding the spectral decomposition of $\Delta= \frac{d^2}{dx^2}$ [closed]

What is the spectral decomposition of the operator $\Delta= \frac{d^2}{dx^2}$ in $(L^{2}(\mathbb R), dx)$? Thanks you in advance
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Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$

Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$, looking at $C_{(2)}[-1,1]$, with $L_2$ norm. I tried to look at a general polynomial $\sum_{i=0}^{98} a_ix^...
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38 views

positive elements and norm

If $A$ is a abelian $C^∗$-algebra and $a,b$ are elements in $A$ such that $0‎≤‎a‎≤‎1,0‎≤‎b‎≤‎1‎‎$ ‎‎ then $0‎≤‎\|a-b \|≤‎1‎$. My problem is:"Does the same hold if $A$ is not abelian?"
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32 views

Exponential Operators

What is wrong ? We have the following identity I try to check the equation, but I get a different answer I have considered the simplest case, in theory I should get two commutators but ... ...
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1answer
40 views

Exponential form for matrices

I'm trying to prove that for two commutative matrices $N$ and $M$, that $e^{N+M}=e^Ne^M$. I wrote using the binomial expansion and commutativity: $$e^{M+N}=\sum_{k=0}^{\infty}\frac{1}{k!}(M+N)^k=\...
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2answers
49 views

Consequence of the polarization identity?

Here is a proof which I do not fully understand. Theorem : Let $H$ be a Hilbert space. A continuous linear map $T : H \rightarrow H$ is self-adjoint (hermitian) if and only if $$\big\langle T(x), ...
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1answer
55 views

What is the dual of the disc algebra viewed as a Banach space?

Let $A$ be the disc algebra, i.e., $A=\{f\in C(\bar{U}):f \text{ is holomorphic in }U\}$, where $U$ is the unit disc in the complex plane. The norm considered is the supremum norm. Are there any ...
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26 views

Validity of inequalities using integrals and absolute value

This question is similar to this one but the only response was pointing out mistakes in the solution. My goal is to determine whether the operator $T: C[0,1] \to C[0,1]$ defined by $Tx = \int_{0}^{t}...
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44 views

Topelitz and matrix operators in $\ell_2$

Let $a,b$ satisfy $|a|,|b|<1$. We then define a vector $y = (\dots,b^2 ,b ,1 ,a ,a^2 ,\dots) \in \ell_ 1 (\mathbb{Z})$ with the ordering $y_0 = 1$. We define a matrix operator $Y$ by $$Yx = \sum _{...
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37 views

Show that an operator is negative

I would show that, the operator $$A = \left(x_{4} \frac{\partial}{\partial x_{1} } -x_{1} \frac{\partial}{\partial x_{2} } \right) \frac{\partial}{\partial x_{3} } $$ is a negative operator on $\...
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28 views

Relation kernels of linear operators

V is a vector space. T and U are two linear operators on V. For finite dimensional, dimker(TU)=dimker(T)+dimker(U) But, what about for infinite case?
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43 views

Property of inversion map on invertible operators

Given $X,Y$ two Banach spaces, I know the set of bounded operators $L(X,Y)$ is Banach with the operator norm $\|A\|=\sup_{\|x\|\leq1}\|Ax\|$. I know the set of bounded operators with bounded inverse $\...
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1answer
33 views

A question on convergence.

If $u_n \rightarrow u$ in $L^p(\Omega)$ and suppose $u_n^{\frac{1}{p-1}}, u^{\frac{1}{p-1}} \in L^p(\Omega) \forall n$ then can it be said that $u_n^{1/p-1}\rightarrow u^{1/p-1}$ in $L^p(\Omega)$?.
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3answers
66 views

Do there exist bounded operators with unbounded inverses?

I have just been introduced to the concept of invertibility for bounded linear operators. Specifically, we defined a bounded operator $A$ to be invertible if there exists a bounded $A^{-1}$ which is ...
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Modified shift operator is compact.

For the operator $$T(\eta_j) = \frac{\eta_{j+1}}{j}$$ on Hilbert Space $H$ where $(\eta_j)$ is a basis. Show it is compact. Can this work? Define $$f = (\eta_j)_{j \geq 1}$$ $$T_N(f) = \left(\...
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1answer
42 views

Spectrum and resolvent of an operator

So for the operator $A:l_2(\Bbb C)\to l_2(\Bbb C)$ defined as: $$A(x_1,x_2,\cdots,x_m,x_{m+1},x_{m+2},\cdots) = (x_1,x_2,\cdots,x_m,0,0,\cdots)$$ We can find the adjoint operator $A^*$ by looking at:...
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71 views

Weak convergence = norm convergence for trace class operators?

Given a (separable) Hilbertspace $H$, I look at the traceclass operators $\mathfrak{S}_1$. I recall the fact that the weak convergence implies norm convergence in the sequence space $\mathcal{l}^1$. ...
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1answer
29 views

Show that the following operator (on a Hilbert space) is continuous.

"Let $\mathcal H$ be a complex Hilbert space and let $y\in\mathcal H.$ Show that the linear transformation $f:\mathcal H\to\mathbb C$ defined by, $f(x)=\langle x,y\rangle$ is continuous." Here ...
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1answer
59 views

Approximation property for Banach space and $l^{p}$

Let's consider a compact operator $T: X \rightarrow l^{p}, 1 \leq p < \infty$. I would like to check, whether it's possible to approximate $T$ by the operators of a finite rank with respect to an ...
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1answer
42 views

Convergence in $L^p$ of product space implies convergence in each space?

Reading a paper by EM Stein (On limits of sequences of Operators, Ann of Math, 1961), the author proves that a certain sequence of functions $F_n(x, t)$, where $x$ belongs to a probability space $(X, \...
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What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds?

Consider the operator $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ acting on a function space $\mathbf{F}(M)$, given by the set of functions $\phi:M\to\mathbb{R}$ whose values go to zero at infinity (at the ...
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24 views

An operator which moves on the boundary

Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis in $H$. Let $E_0$ be a countable subset of $E$ and $p$ be the projection onto the space generated by $E_0$. Let $\{\zeta_n\}...
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1answer
28 views

Show that the following operator is not a surjection.

"Take $X=C([0,1])$ with the uniform norm, $\|f\|=\sup_{x\in[0,1]}|f(x)|$, and define the operator $T:X\to X$ by, $$T(f)(x)=f(x)-\int_0^1f(s)ds$$ Show that $T$ is not a surjection". Here is what ...
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101 views

Find the norm of the following operator.

Take $X=C([0,1])$ with the uniform norm, $\|f\|=\sup_{x\in[0,1]}|f(x)|$, and define the operator $T:X\to X$ by, $$T(f)(x)=f(x)-\int_0^1f(s)ds$$ Find $\|T\|$. I was hoping to solve this problem ...
2
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1answer
40 views

A system of equations

Let $H$ be a non-separable Hilbert space. Assume $E$ is an orthonormal basis in $H$. Let $E_0=\{e_n\}$ be a countable subset of $E$ and let $\{\zeta_n\}$ be a bounded sequence in $H$. Let $E_1$ be a ...
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64 views

Norm of the inverse of a map $\ell^2\to\ell^2$

Let $Au_i=u_{i+1}-(2-\beta)u_i+u_{i-1}$ whith $u\in \ell^2=\{(u_i)_{i\in \mathbb Z}, u_i\in \mathbb R:\sum_{i\in \mathbb Z}u^2_i<+\infty\}; \beta>0$. How to compute $||A^{-1}||$ or estimate it? ...
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36 views

Is the set of linear combinations dense in the set of the dual space of $l_p$?

Good day, Right now I'm working with the book "Functional Analysis" by Bachman and Narici, it is available on Google Books, see https://books.google.de/books?id=wCHtLumoGY4C&printsec=frontcover&...
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21 views

Fréchet differentiability of Nemyckij operator defined on $L^2$

I have been told the following. Suppose $\Omega\subseteq\mathbb{R}^n$ is a bounded borel set, $f$ is Carathéodory function on $\Omega\times\mathbb{R}=\{(x,s):x\in\Omega,s\in\mathbb{R}\}$, $f_s$, ...
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58 views

Showing that the operator is bounded and find its norm.

I have this operator $T: L^p(0,\infty)\rightarrow L^p(0,\infty)$, $1<p<\infty$ : $(Tf)(x)=1/x\int_0^xf(t)dt$. I am supposed to show that it is bounded and fint its norm. I had an idea that ...
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2answers
38 views

Integral of an operator

In quantum mechanics we know that if $q$ corresponds to a complete set of parameters characterizing a quantum system, then the state vectors $|q\rangle$ satisfy the following identity: $$\int |q\...
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15 views

convergence rate of forward backward operator splitting algorithms

I am looking for some latest material on convergence rate of the basic forward backward operator splitting algorithm. After googling, I found the following: http://epubs.siam.org/doi/abs/10.1137/...
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1answer
57 views

Real-Valued Symmetric Square Matrices and Min-Max

A real-valued symmetric square matrix is called positive definite if $(x,Ax)>0$ for all $x\neq0,$ where $(.,.)$ represents the scalar product. For a positive definite matrix determine $$\max\left\{...
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$A^2=A $ prove $A$ is hermite matrix

Let matrix $A\in M_n(\mathbb{C})$ satisfying $A^2=A$ , for every $n\times 1$ vector $x$ we have $|Ax|\le|x|$ where $| |$ denotes the usual norm of vector , prove $A$ is a hermite matrix.
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30 views

Derivation of an integral equation

I have the following system $$\frac{d}{dx}\left(a(x)\frac{du}{dx}\right)=f, \text{ for } x \in (0,1)$$ with boundary conditions $u_x(0)=0$ and $u(1)=0$. For $a(x)>0$, and $b(x)=\frac{1}{a(x)}$, I ...
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39 views

An operator satisfying in a sequence of equations

Assume that $H$ is a non-separable Hilbert space. Let $\{\eta_n\}$ be an arbitrary sequence in $H$. Let $\{\zeta_n\}$ be a sequence in $H$ which forms a linearly independent set. Does there exist ...
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20 views

A translation invariant sigma algebra in $B(H)$

Assume that $H$ is a non-separable Hilbert space. Let $s_0$ be the family of all basic neighborhoods in the strong operator topology. We denote $M_s$ by the sigma algebra generated by $s_0$. ...
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58 views

Eigenvalues of an integral operator

The following operator is defined on $L_2(0,1)$: $$Kf(t)=\int_0^1|s-t|f(s)ds$$ I am wondering how I can calculate the eigenvalues and eigenfunctions of such an operator. I start with $\int_0^1|s-t|f(...
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1answer
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Norm of $T^n$, where $Tf(x,y) = \begin{cases}f(x+y/b,y), &0<x<1-y/b,\\1/2f(x+y/b-1,y),& 1-y/b<x<1.\end{cases}$

Let $0 < a < b$ and $T\colon L^\infty((0,1)\times (a,b)) \to L^\infty((0,1)\times (a,b))$ be the operator defined by $$Tf(x,y) = \begin{cases}f(x+\frac yb,y), &0<x<1-\frac yb,\\\frac ...
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1answer
30 views

adjoint operator of the partial trace map

Could someone explain to me, what is the adjoint map of the partial trace map the (tensored with the identity map), or why does the following equality hold? $Tr(C_A\cdot Tr_{B} D_{AB})=Tr((C_A\otimes ...
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18 views

Inverse problem of covariance matrix – diagonalization of Hermitian operator

I have understood the two things respectively: 1. Use a set of observations to construct a covariance matrix, and then compute the eigenvectors of the matrix. 2. The diagonalization the Hermitian ...
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1answer
51 views

Closure of a differential operator

Consider $A:\mathcal{D}(A)\subset L^{2}[0,1]\to L^{2}[0,1]$ given by $$A:=-\frac{d^{2}}{dx^{2}},\qquad\mathcal{D}(A):=C^{2}_{0}(0,1)$$ Now, I assume that the closure of $A$ is its extension defined ...
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$\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is a stable C*-algebra

Let $B$ be a C*-algebra. I want to prove that $\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is stable, that is, $\mathcal{K} \otimes B$ is isomorphic to $B$ as C*-...
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1answer
28 views

Is there any standard name for this theorem about extension of bounded linear operators in normed spaces without changing the norm?

Let $X$ and $Y$ be normed spaces, both real or both complex; let, in addition, $Y$ be a Banach space; let $V$ be a (vector) subspace of $X$; let $T \colon V \to Y$ be a bounded linear operator; and ...
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2answers
49 views

In a normed vector space X: $x_n \to x$ weakly iff $d(x_n) \to d(x) ~\forall d \in D$, $D$ dense in $X^*$

Good day, I have the following task: Let $(x_n)_{n \in \mathbb{N}}$ be a sequence in the normed vector space $(X, || \cdot || )$ and let $x \in X$. Show that the following are equivalent: (i) $x_n$ ...
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3answers
51 views

Does it follow that the hermetian part of a matrix is positive definite, that the matrix itself is invertible? [duplicate]

I came across this in a paper and I was wondering whether it is true. We have a complex matrix $M$ such that $(M+M^H)$ is positive definite. Now, it is clear that $(M+M^H)$ is invertible, but does ...
2
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2answers
89 views

How to show that a operator is (not) self-adjoint? [closed]

In order to prove that an operator is self-adjoint or not, what should I do? For example, how can I show that the following operator is self-adjoint? $K: C[0,1]\to C[0,1]$ with $$[Kf](x)=\...
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2answers
12 views

Proving equivalence of operators imply equivalence of measures

Let $A:L^2([0,1],\mu)\to L^2([0,1],\nu)$ an unitary operator. Prove that $$d\mu=\rho(x) d\nu$$ for some $L^1(\mu) \ni \rho(x) >0 (\mu\text{ a.e})$ I thought maybe saying $$\int_{[0,1]}|f(x)|^2d\...
5
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1answer
101 views

Connection of Fourier's work with Fredholm's

Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with ...
0
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0answers
52 views

Linear operator between $l^\infty$ and $l^2$

Let $T:\mathcal{l}^{\infty}(\mathbb{R})\to\mathcal{l}^2 (\mathbb{R})$ be given by $$ T\left((x_n)_{n\in \mathbb{N}}\right) \colon= \left(\dfrac{1}{2^n} x_{2^n}\right)_{n\in \mathbb{N}}.$$ Find $\...