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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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 ...
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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 ...
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97 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 ...
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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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35 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 ...
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20 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 ...
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13 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: ...
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1answer
45 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 ...
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29 views

$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 $s_0$. ...
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1answer
57 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 ...
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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 ...
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1answer
29 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
50 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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76 views

$\mathcal{H}_B$ 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 ...
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1answer
26 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; ...
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2answers
45 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
50 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 ...
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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 ...
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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 ...
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100 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 ...
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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 ...
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40 views

Is there a mistake in the solution given to this exercise?(compact linear operators)

The exewrcise is: Let $\mathcal{H}$ be an infinite-dimensional Hilbert space, with an orthonormal basis $\{e_n\}$ and let $T \in \mathcal{B}(\mathcal{H})$. Show that if T is compact then ...
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45 views

Example of a densly defined positive self adjoint unbounded operator.

I know example of a densely defined positive self-adjont unbounded operator with discrete spectrum. What is the example of a self-adjoint positive unbound operator with continuous spectrum?
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52 views

An strange operator in B(H)

Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis for $H$. Let $E_0$ be a countable subset of $E$ and $\{\delta_i\}_1^{\infty}$ be a bounded sequence of $(0,\infty)$. For ...
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Is it possible to characterize completeness of a normed vector space by convergence of Neumann series?

If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? ...
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33 views

About the spectral radius of an integral operator

My question is given at the end of the explanation. Let $K\in{}C([a,b]^{2},\mathbb{R})$ and consider the operator $H:C([a,b],\mathbb{R})\to{}C([a,b],\mathbb{R})$ defined by ...
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34 views

Boundedness of singular integral operators on $L^{p}$ spaces

Let $\Omega \in L^{1}(S^{d-1})$ have mean zero. Prove that, if the operator $T_{\Omega}: L^{p} \rightarrow L^{q}$ given by $T_{\Omega}f(x) $:= p.v. $\int_{\mathbb{R}^{d}} \frac{\Omega ...
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2answers
77 views

Polar decomposition of Bounded Normal Operator on Hilbert Space

It is well known that if $T$ is a bounded linear operator on a infinite dimensional Hilbert space $H$ then there exists unique partial isometry $U$ such that $T=U \vert T \vert$,where $\vert T \vert ...
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51 views

Laplace-Beltrami operator

I'm interesting in the Laplace-Beltrami operator on a sphere, more precisely its spectral properties including the spectral function, etc. So if someone can give me some references that treats this ...
2
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1answer
87 views

Spectrum of nonnegative operator

Let $A$ be a bounded, nonnegative operator on a complex Hilbert space $H$. Prove that the spectrum $$\sigma(A)\subset[0,+\infty].$$ We say that an operator $A$ is nonnegative if it is self adjoint and ...
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1answer
28 views

Uncountable linearly independent set of vectors implies the space is not separable? + A question on self adjoint operators

Let $\mathcal H$ be a hilbert space and let $A\subset \mathcal H$ be an uncountable set of linearly independent vectors. Does this imply $\mathcal H$ is not separable? If $A$ was a set of orthonormal ...
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1answer
31 views

Adjoint or Formal adjoint? Confusion regarding gradient and laplacian

I have recently become a bit confused with the distinction of adjoint and formal adjoint. I looked at some old threads here but did not really find the explanation I am looking for. So, I know that ...
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1answer
27 views

Let $T:\ell_1 \to c_0$ be linear operator defined as $x_n \to \sum_{k\geqslant n} x_k$. Then $T \in B(\ell_1, c_0)$.

I'm solving some exercises for my Functional Analysis Exam. Here is one on which I am stuck: Let $T: l_1 \to c_0$ be linear operator defined as $x_n \to \sum_{k\geqslant n} x_k$. Show that $T \in ...
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22 views

Orthogonal, Normal, and Self-Adjoint operators

Let be $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ the operator given by $T=3ref_{x_2=2x1}$, where $ref_{x_2=2x1}$ is the reflection on the line $x_2=2x_1$. Define the operator $U:{(\mathbb{R}^2)}^n ...
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51 views

Is there a way to find the operator norm in this case?

If $k:[a,b]\times[a,b]\rightarrow \mathbb{R}$ is in $L^2([a,b]\times[a,b])$, we can show that the linear operator: $T_k: L[a,b]\rightarrow L[a,b]$, given by ...
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1answer
33 views

Laplacian operator defined in $L^2(0,+\infty)$

I have this problem that doesn't seem too simple to me. Let be $-\Delta$ the laplacian operator with domain in $C^\infty_0(0,+\infty)\subseteq L^2(0,\infty)$ I want to prove the following: ...
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1answer
63 views

Orthogonal projection

Given the Hilbert space $H=\mathbb{L}^2 ([0,1])$ endowed with the canonical scalar product $\langle f,g \rangle = \int_0^1 f(x)g(x)dx$, as well as the orthogonal projection $p_n:H\mapsto H$ defined ...
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34 views

Spectral properties of operator over $L^2(0,1)$

Let be $I=[0,1]$ the unit interval and define the operator $$(Af)(x)=\int_0^x f(t)dt $$ with domain $C_0^{\infty}(I)\subseteq L^2(I)$. I want to show the following: $A$ is bounded and compact; ...
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2answers
59 views

Is intersection of a dense subspace and a closed subspace of a Hilbert space also Dense?

I have a Hilbert space $H$ and a closed operator $T$ defined on its domain $D(T)$ which is dense in H. Also $M = \text{range} \ T^n$, for some $n$, is given to be closed. Consider the restriction of ...
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1answer
32 views

Gaussian integral for a vector and a function - how to evaluate

My problem concerns evaluation of a Gaussian integral. Let there be a real vector $\mathbf{v}$ and a matrix $\mathbf{A}$. I would like to know the result of the following integral: $$ ...
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1answer
37 views

Operator norm: Show that $\|A\|=\sup\{ |f(Ax)| : x \in X, \|x\| \leq 1 , f \in Y^*, \|f\| \leq 1 \}$

Good day, As stated in the title, I have to show that $\|A\|=\sup\{ |f(Ax)| : x \in X, \|x\|_X \leq 1 , f \in Y^*, \|f\| \leq 1 \}$ where $\| \cdot \|$ is the operator norm, i.e. for $X,Y$ vector ...
0
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0answers
30 views

Ladder (recurrence) operators for Hermite polynomials?

Generally, Hermite polynomials can be described using the Rodrigues formula: $$ H_n(x) = (-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2} $$ And the first few polynomials (for $n = 0,1,2,3,4,5...$) are well ...
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1answer
39 views

Adjoint of a matrix vs adjoint operator

I am having some confusion over how I would attack a proof of the properties of matrix adjoints. Here is an example: Let $A,B\in M_{n\times n}(\mathbb{F})$ with adjoints $A^*$ and $B^*$. Prove ...