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 matrix $[S]$ associated to operator is such that $A |\zeta|^2\leq s_{ij}(x) \zeta_i \zeta_j\leq B |\zeta|^2$.

Let us consider $N\times N$ matrix $[S]$ associated to operator $S:V\rightarrow V$ where $V$ is a Hilbert space; $S$ is linear, bounded, invertible, positive and self-adjoint. Prove that $[S]$ is ...
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39 views

Spectral theorem for compact normal operators

Let $H$ be a Hilbert space and $A$ a compact normal operator from $H$ to $H$. How to show that its eigenspaces produce the space? I can show it for self-adjoint operators and by setting ...
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1answer
43 views

Norm of the quotient map for a normed space [duplicate]

Let $X$ be a normed space and $F$ a closed subspace. On $X/F$ let us take the quotient norm $||[x]|| = \inf_{y \in F} ||x - y||$. Consider the quotient $q : X \rightarrow X/F$. I can see that, if ...
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2answers
47 views

How to find the poles of a green function?

I am trying to construct a green function for $y''+\alpha^2u=f(x), u(0)=u(1), u'(0)=u'(1)$. For that I am trying to follow the procedure described here:(Construct the Green s function for the ...
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23 views

Showing that an operator is bijective

Assume that $ A $ generates a contraction semigroup on a Hilbert space $ X $, and B is a bounded linear operator on $ X $. I want to show that $ A + B - 2|| B ||I $ with the domain equal to the domain ...
4
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2answers
40 views

An invertible sparse matrix?

I'm not entirely certain about how to tackle this problem.... I hope you ladies and gents can help :) If $M\in M_{n\times n}(\mathbb{R})$ be such that every row has precisely tow non-zero entries, ...
4
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1answer
36 views

$\overline{\mathrm{Im} (T^*T)} = \overline{\mathrm{Im} T^*}$

I need to prove that in a Hilbert space, $\overline{\mathrm{Im}(T^*T)} = \overline{\mathrm{Im}T^*}$. I have already shown that $\ker (T^*) = (\mathrm{Im} T)^\perp$ and have so far concluded that ...
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19 views

Origins of the Cesaro Operator

I am wondering when the Cesaro Operator was first studied. I can find an article from 1965 but I'm wondering if there are any previous ones.
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2answers
36 views

Inverse of $I +T^*T$

I am trying to show that the inverse of the operator $I +T^*T$ exists. What I have been trying to do is trial and error taking inverses of $T$ and $T^*$ but to no avail.
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1answer
22 views

Connection between Stinespring's factorization theorem and the spectral theorem for bounded operators

I know at least 2 versions of a Spectral theorem for operators, one of them is the following Theorem: Let H be a separable complex Hilbert space, $A\in L(H)$ self-adjoint ($L(H)$ are the bounded ...
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3answers
36 views

$A$ and $B$ are bounded linear operators from the normed linear space $X$ to itself. If $AB$ is invertible are $A$ and $B$ invertible?

I think I understand the proof for square matrices, such that $(AB)^{-1}=B^{-1}A^{-1}$, but I'm not sure if I can just say the same for the bounded linear operators A and B. Does the existence of ...
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31 views

Why is the Calkin algebra purely infinite?

I tried using the fact that in a simple unital $C^*$-algebra, $\mathcal{A}$, purely infinite is equivalent to the following: If $x\in\mathcal{A}$ is non-zero, then there exists $a,b\in\mathcal{A}$ ...
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0answers
45 views

Prove that norm of Yosida approximations blows up

I need an help with the following exercise about Yosida approximations. Let's fix the notation: let $A$ be an unbounded linear operator define on a dense domain $D(A)\subset X$ of a Banach space $X$. ...
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1answer
14 views

Show that an operator is closable

Let $H=\mathcal{L}^2(\mathbb R^2,dxdy)$ and let $A$ the operator defined by: $$ A[f](x,y)=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}+i(y\frac{\partial f}{\partial ...
9
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1answer
133 views

Spectral theorem for a pair of commuting operators

Let $H$ be Hilbert space and $A$, $B$ - self-adjoint (bounded or unbounded) operators on $H$. According to spectral theorem for every bounded Borel function $f: \mathbb{R}\to \mathbb{R}$ we have ...
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43 views

Show $\lim_{n\to\infty}(Lu_n) = L(\lim_{n\to\infty} u_n)$

Suppose $\{u_n\}$ is a convergent sequence in Hilbert space $H$ and $L$ is a bounded (continuous) linear operator on $H$. Use the definition of convergence to show that $\lim_{n\to\infty}(Lu_n) = ...
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41 views

Find eigenfunctions of the integral operator with kernel $\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny)$

Find the eigenvalue and eigenfunctions of the integral operator $Ku=\int_0^\pi k(x,y)u(y)dy$. $k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$. This is how I ...
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42 views

Derivation of perturbation series

I'm a little bit confused about the derivation of the perturbation series. I know from my quantum mechanics course that for a perturbed operator, eigenvalue is a series that is depend on the ...
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1answer
28 views

Is it necessary to use the Hahn-Banach theorem to show that $(X/M)^*\simeq M^\perp$?

Let $X$ be a Banach space with dual space $X^*$, and let $M$ be a closed subspace of $X$. Then $M^\perp=\{x^*\in X^*: x*(m)=0 \text{ for all } m\in M\}$ is a closed subspace in $X^*$. I read the ...
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3answers
91 views

looking for help with a trace/norm inequality

I'm trying to understand a derivation that seems to claim that $\left\vert\text{Tr}\left[\rho U^\dagger\left[U,O\right]\right]\right\vert\leq\|\left[U,O\right]\|$, where $\rho$ is Hermitian and has ...
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1answer
42 views

When can we exchange the trace and an integral/limit/derivative?

For a trace class operator $A$ (acting on a Hilbert space), that is parameterised by a real variable $x$, what are the conditions for the following to hold? $$ \mathrm{tr} \int_a^b A(x) \, dx = ...
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1answer
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symetric closed operator and extension [closed]

i have this question let A a symetric closed operator let pose that A have a self adjoint extension is possible that A has an extension such that closure A can't have a self adjoint extension
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1answer
22 views

What can one assume about $T^*$ when showing that $T$ is normal?

Consider a continuous and linear operator $T$ such that $$ T : l^2 \to l^2 $$ where $(a_n) \mapsto (\alpha_n a_n)$ Moreover $(\alpha_n)$ is a sequence of complex numbers that converges to zero. Now, ...
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22 views

Norm of operator matrix

I'm having trouble with the following: suppose H is a Hilbert space and $f_{i, j}, g_{i, j} : H \rightarrow H$, $1 \leq i, j \leq n$ are bounded operators. Then we have operators $(f_{i, j}) , (g_{i, ...
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0answers
29 views

Multiplication operators are sectorial

Consider the multiplication operator $M_a$ on $L^p(M)$, where $M$ is a Riemannian manifold, and $a$ is a non-negative function. An operator $A$ is said to be sectorial if there exists $\theta \in (0, ...
2
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1answer
92 views

Showing that an operator generates a contraction semigroup

Let $A$ be the infinitesimal generator of a contraction semigroup $(T(t))_{t\ge 0}$ on the Hilbert space $X$, and $D\in\mathcal{L}(X)$. I want to show that the operator $A+D-2\|D\|I$ with domain ...
2
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0answers
14 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if $A$ and $B$ are two $n \times n$ Hermitian matrices, and $[A,B]=C$. I'd like a ...
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1answer
67 views

Why has the Stein operator for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$?

My Question: Why has the Stein operator $\mathcal A$ for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ...
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2answers
25 views

A relation between the domain of $A$ and the domain of $\bar A$

Let $A$ be an operator: $$ A:D(A)\to R(A) $$ where $D(A)$ and $R(A)$ are respectively the domain and the range of $A$ and they are subspaces of a Hilbert spcae $(H,\|\|)$. Suppose that $A$ is a ...
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73 views

proving that $\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$

In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that ...
2
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1answer
57 views

Powers of compact operators

Consider a Hilbert space $H$ and a compact self-adjoint operator $T : H \to H$. I want to prove that all positive powers (especially fractional powers) of $T$ are compact. From the spectral theorem, I ...
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1answer
38 views

Position operator is self adjoint

Let $H=L^2(\Bbb{R})$ with the linear (unbounded) operator $P(f)(x)=x\cdot f(x)$ for each $x\in\mathbb{R}$. Have a look at the following domain: $$D(P)=\{f\in ...
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2answers
228 views

Proving that $\text{ri rge}\,A=\text{ri conv rge}\,A$

"If $A:\mathbb R^n\rightrightarrows\mathbb R^n$ is maximal monotone,then $\text{ri rge}\,A$ is convex". This is a proposition in auslender's book about the asymptotic cones. We can prove that ...
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1answer
44 views

How to use logical conjunction properly

On this website in equation (20) they use $$ d \, S = a \, d \, u \land d \, v $$ I have learned that $\land$ is the truth-functional operator of logical conjunction and that such logical operators ...
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1answer
39 views

Clarification on some definitions in Operator Theory

I'm trying to read this paper http://arxiv.org/abs/1206.3325 , but I'm having a lot of difficulty making sense of two phrases. The setting is $L_2(\mathbb R^d)$. i) He mentions that for a function ...
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1answer
144 views

Proving that T(t)x is in the domain

$(T(t))_{t\ge 0}$ is a $C_{0}$-semigroup on a Banach space $X$ with generator $A:D(A)\subset X\to X$. For $k\ge 2$, define $$D(A^{k}):=\{x\in D(A^{k-1})|A^{k-1}x\in D(A)\}$$ I want to show that for ...
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2answers
69 views

Does there exists an operator with these properties?

Consider with $(\Omega,\Sigma,\mu)$ a $\sigma$-additive measure space. Is there a linear operator $P \neq 0$ $$P : L^1(\mu) \to L^1(\mu) $$ which fulfills $$ \|Pf \| \leq \|f\|,$$ $$ f\geq0 ...
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1answer
49 views

Extension of bounded operators between norm spaces

Let $X$ and $Y$ be two Banach Spaces and $X_1$ be a subspace of $X$. If $T$ is a bounded linear operator from $X_1\to Y$, then, this is an extension of $T$ from $X\to Y$ such that $\|T\|_{X}\le ...
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1answer
31 views

Does any continuous map between two Banach spaces over a nonarchimedian field is a closed map?

Does any continuous map between two Banach spaces is a closed map? It seems to me that it is true. Let $f:V \rightarrow W$ be a map, where $V$, $W$ are (infinite dimensional) Banach space over a a ...
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34 views

Hamiltonian: Invariant Core

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote its evolution by: ...
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1answer
28 views

Reducing Spaces: Hamiltonian

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a projection: $$P\in\mathcal{B}(\mathcal{H}):\quad P^2=P=P^*$$ Then one has: ...
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1answer
20 views

Selfadjoint Operators: Weak Convergence

Given a Hilbert space $\mathcal{H}$. Consider a selfadjoint operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Regard a sequence: ...
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1answer
38 views

Laplace transform, Bochner integral

I have a quesition about linear operators on a Banach space. Let $B$ be a real Banach space. $(T_{t})_{t>0}$ is called strongly continuous contraction semigroup on $B$ if For all $t>0$, ...
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23 views

Integral kernels of self-adjoint operators

If the integral kernel $k(x, y)$ of an operator $T : C^\infty_c(M) \to \mathcal{D}'(M)$ is symmetric ($M$ is a compact manifold), then the operator $T$ is symmetric. Is the converse true? That is, ...
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19 views

Operator norm of symmetric Matrix in Hilbert Space with Hermitian Inner Product

Assume we have a postive definite real matrix $P$ and we define an inner product on a finite dimensional hilbert space $\langle x, y \rangle = x^\top P y$ and clearly the induced norm is $|| x || = ...
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1answer
34 views

Bounded linear functionals over smooth maps of a compact interval

I have two questions regarding the topological dual of the space $E = \mathcal{C}^\infty([0; 1])$ of infinitely continuously differentiable functions over the closed interval $[0; 1]$ equipped with ...
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29 views

Is there a pseudocontractive mapping that is not strictly pseudocontractive?

Given a Hilbert space $H$, a mapping $T:H\rightarrow H$ is said to be pseudocontractive if $$\|Tx-Ty\|^2\leq \|x-y\|^2+\|(x-Tx)-(y-Ty)\|^2\,\,\, \forall x,y\in H,$$ and it is strictly ...
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1answer
31 views

Operator - Exponential form

It is well known that for every unitary operator $\hat U$ an exponential of the form $$ \hat U = e^{i\hat H} $$ exists ($\hat H$ is hermitian). But I can only prove it the other way round: $$ ...
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1answer
30 views

Weak operator limit of projections in $B(H)$

Let $H$ be infinite dimensional and $\cal P$ be the set of all projections in $B(H)$. Show that $\cal P$ is weak operator dense in $(B(H))^+_{\|.\|\leq 1}$, the set of positive operators in the unit ...
2
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0answers
38 views

Quartic operator definition

What is a quartic operator? I googled it but found only some articles which use that term whitout giving a definition (I found that term while studying 2D Ising model, and the use of some ...