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 the definition of distinguished unit vector

What is the definition of distinguished unit vector? I guess it is the identity element,is it right?
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33 views

If there exists a bounded operator $T:X \to Y$ with $T^{-1}$ bounded then X is Banach iff Y is Banach

I have been asked to show that if there exists a bounded operator $T:X \to Y$ with $T^{-1}$ bounded then X is Banach iff Y is Banach. I have shown it for $T$ a linear operator. But I can't use the ...
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58 views

Show that there exists a certain operator in $L(H)$ where $H$ is a separable Hilbert Space.

Given a separable Hilbert Space $H$ and $\sum_{n=1}^{\infty} f_n$ an absolutely convergent series in $H$, I need to show that there is an operator $A \in L(H)$ such that $A(e_n)=f_n$, where $(e_n)...
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1answer
24 views

Equality for functions in $H^2(\mathbb{R})$

I recently stumbled on the following equality: $$ \| (-\frac{d^2}{dx^2} + 1)^{1/2}g\|_{L^2} = \| g\|_{H^1}$$ for $g \in H^2(\mathbb{R})$. I tried to deduce the equality but failed (since I don't ...
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1answer
29 views

adjoint of operator?

let $H=L^2(0,1)$ (Hilbert space with usual scalar product )and the operator $A$ defined by : $D(A)=\{u\in C^1[0,1]:u(0)=\lambda u(1)\}$ where $\lambda\in\mathbb C$ and $Au=iu'$ my questions is : ...
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1answer
21 views

Operator norm and continuity

I've read in the solution of an exercise: "$T$ has a finite norm, thus $T$ is continuous". We are in a normed vector space $(V,||.||)$ and $T$ is a linear selfmap over the vector space $V$. The ...
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36 views

a proof about closable operator

I am self-studying the chapter of closed and closable operators. I have the following problem which I cannot find its proof. Let $A$ be a closable operator and denote by $B$ a closed extension. We ...
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353 views

Cyclic vector in $\ell^2(\mathbb{Z})$ space

Suppose we look at $\ell^2(\mathbb{Z})$, which contains vectors $c=(\dots,c_{-2},c_{-1},c_0,c_1,\dots)$ with $\sum|c(n)|^2<\infty$. Define the right-translation operator by $$R:\{c(n)\}\mapsto \{c(...
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1answer
67 views

Prove: Expectation value is the weighted average

How do we mathematically prove that the expectation value of an operator is the weighted average, $$ \langle\hat{A}\rangle=\langle \psi|\hat{A}|\psi\rangle=\sum_{n}a_{n}P(a_{n}) \space \space \space? ...
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52 views

Isolated Eigenvalues on the Extensions.

I asked this question on Mathoverflow http://mathoverflow.net/questions/226484/isolated-eigenvalue-of-t-is-also-an-isolated-eigenvalue-of-overlinet and because of the comments apparently the answer ...
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42 views

Find the Left Inverse of an Operator

Suppose I have the following operator defined on the infinite line $L = \frac{\partial}{\partial t} + \lambda t$ Now, I can find the right inverse of this operator, $G(t,t_0)$, by solving the ...
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41 views

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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1answer
46 views

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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45 views

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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1answer
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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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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1answer
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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1answer
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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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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47 views

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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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
60 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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92 views

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 ...
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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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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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22 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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1answer
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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1answer
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 $s_0$. ...
3
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1answer
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
22 views

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 ...