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 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 ...
4
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37 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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0answers
36 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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0answers
29 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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0answers
38 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
25 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
62 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 ...
4
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1answer
39 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
15 views

symetric closed operator and extension [on hold]

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
20 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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0answers
20 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, ...
2
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0answers
27 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
59 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
10 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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0answers
21 views

Hamiltonian: Scattering Spaces

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a family of projections: $$1(r)^2=1(r)=1(r)^*\quad(r\geq0)$$ Denote for shorthand: ...
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0answers
42 views
+50

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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0answers
20 views

Any injective *-homomorphism between finite dimensional C*-algebras is an isometry

I believe the answer is true, but I need some guidance with a proof. Does anyone have a hint toward either a proof or counterexample? Thanks
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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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2answers
60 views

Theorem 6.4.1 Auslender Asymptotic cones and functions in optimization and variational inequalities

In proof of Theorem 6.4.1, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that $\epsilon^{-1}(C-\text{rge}\,A)\subset\text{aff}\,(C-C)$, that I can't verify ...
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1answer
46 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
25 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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1answer
153 views
+50

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 ...
2
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1answer
41 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
35 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
120 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
56 views
+50

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
44 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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0answers
22 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 ...
2
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0answers
29 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
18 views

Reducing Subspaces: 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
19 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
36 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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0answers
18 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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0answers
15 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
29 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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0answers
22 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
24 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
26 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 ...
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30 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 ...
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0answers
37 views

Does a Plancherel Style Theorem for the Hardy Space $\mathbb{H}^2$ on the Unit Circle Exist?

I am working on a problem regarding Toeplitz operators, and it involves trying to prove $\mathbb{H}^2$ boundedness of the operator (defined in terms of its Fourier coefficients). Now normally when I ...
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1answer
67 views

Let we have the following exercise [closed]

How can I solve the following exercise Let $T:C[0,1]\rightarrow C[0,1]$ be defined by $$Tx(t)=y(t)=\int_0^t x(\tau)d\tau.$$ Find $\mathscr R(T)$ and $T^{-1}:\mathscr R(T)\rightarrow C[0,1]$. Is ...
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2answers
37 views

What is the range of the operator $T$ I mean I want to determine $R(T)$

Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
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2answers
86 views

I want some help in functional analysis [closed]

I want sone help in functional analysis : $1)$ consider the vector space $X$ of all real -valued functions which are defined on $R$ and have derivatives of all orders everywhere on $R$ define ...
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1answer
19 views

Two self-adjoint operators with the same eigenvalues and eigenfunctions

How to show two self-adjoint operators (unbounded) on a Hilbert space with the same eigenvalues and eigenfunctions are the same.
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1answer
27 views

Unitary element in a C*-algebra

Suppose $\Bbb T$ is the unit circle and $M$ is the C*-algebra of all $2\times 2$ complex matrices. Consider the C*-algebra $A: = C(\Bbb T, M)$. Let $E$ and $F$ be the projections in $A$ given by ...
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1answer
31 views

Are symmetric matrices necessarily positive-definite / positive semi-definite?

I am trying to prove this just to be clear about this but I don't have enough conditions to force this idea to be true, so I doubt it is. Are symmetric matrices always at least positive ...
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1answer
79 views

Some question about linear operator on normed space [closed]

$1)$ let $X$ and $Y$ be normed space , show that a linear operator $T:X\rightarrow Y$ is bounded if and only if $T$ maps bounded sets in $X$ into bounded sets in $Y$ $2)$show that the operator ...
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0answers
18 views

Singular Spectrum: Techniques?

Given a Hilbert space $\mathcal{H}$. Let the Lebesgue measure be $\lambda$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote its spectral measure by: ...
2
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1answer
36 views

If I want to prove that $M^{\perp}$is a closed

If I want to prove that $M^{\perp}$is a closed Can I say because it is the inverse image of $0$ by continuos function ( projection operator )