Questions tagged [operator-theory]

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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7 views

Minimax type principle for a self-adjoint operator acting on a Hilbert space

Let $T\in\mathcal{B}(\mathcal{H})$ be a self-adjoint operator acting on a Hilbert space $\mathcal{H}$. Suppose $k\in\mathbb{N}$. Define $$\lambda_k(T)=\sup\left\{\lambda_k(V^*TV):V:\mathbb{C}^k\...
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Prove: $n^{-1} tr(DG^{-2}-D_0G_0^{-2}) \to 0$ if $D_0,G_0\in R^{n \times n}$ have bounded spectra, and $\|D-D_0\|,\|G - G_0\| \to 0$ as $n \to \infty$

Let $n > 0$ be a large integer and let $D,D_0,G,G_0$ be $n \times n$ psd matrices such that $\|D-D_0\|_{op} = o_n(1)$ and $\|G-G_0\|_{op} = o_n(1)$. $\|D_0\|_{op},\|G_0\|_{op},\|G_0^{-1}\|_{op} = ...
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Operators on Hilbert Spaces

Question: Consider $A \in \mathcal{L}(\mathcal{H})$, where $\mathcal{H}$ is a Hilbert space. Show that $\|A\|=\sup_{u,v \neq 0} \frac{|\langle u,Av \rangle|}{\|u\|\|v\|}$. Remark: Here $A$ is an ...
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Models of Mikusinksi's operational calculus

One construction I'm particularly fond of is Mikusinki's operational calculus - that is, the field of convolution fractions of functions from $\mathbb{R}^+ \to \mathbb{R}$. To construct this, we ...
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Any invariant translation operator T can be represented by a multiplication operator on the Fourier transforms.

In Harmonic Analysis by Stein E. Why $T(e^{2\pi i\cdot \xi})=a(\xi)e^{2\pi ix\cdot \xi}$? I don't see how to deduce that.
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116 views

What can we say when $[A,B]=A$? [closed]

Let $A,B \in \mathbb{C}_{n\times n}$ and let $[A,B]$ be their commutator. What can we say when $$ [A,B]=A \quad ? $$ It feels like this is something interesting to study (specially in physics). Does ...
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Delay differential equations via semigroup [closed]

Where can I find a good reference on the approach via semigroups of linear bounded operators to delay differential equations?
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14 views

Construction of a regularizer for an operator written as a product of positive-definite symmetric matrix and a positive selfadjoint operator

Let $H$ be a Hilbert space, $M$ a $3\times 3$ real constant symmetric positive matrix and $A:H^3\to H^3$ a bounded positive selfadjoint operator. In the paper "Spectrum of the Product of ...
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Lower bound on norm of the difference of two matrix [closed]

Suppose $A$ and $B$ are two positive semi-definite matrices, is it possible to prove or falsify that $\Vert A - B \Vert_2 \geq \lambda_{\mathrm{min}}(A) - \lambda_{\mathrm{min}}(B)$, where $\lambda_{\...
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35 views

Is the following operator trace class?

Let $H$ Hilbert with orthonormal basis $\{e_k\}$, $B \colon H \to H$ linear and bounded, invertible. $Q \colon H \to H$ linear operator, not trace class, i.e. $$tr Q =\sum_{k \in \mathbb{N}} \langle ...
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64 views

trace of inverse operator

Let $H$ be an RPK-Hilbert space and $K:X\times X\rightarrow \mathbb{R} $ be the reproducing Kernel s.t. $K$ is bounded by $1$. For some Probability Space $(X, \nu)$ It is assumed that all $f \in H$ ...
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41 views

When the image of operator be subset of $l_p$

Let $f:l_{\infty} \rightarrow l_{\infty}$ defined by $f(x_1,x_2,...)=(x_1,\frac{1}{2}x_2,...,\frac{1}{2}x_n,...)$. Then we need to (1) prove that $f$ is continuous operator and (2) find $p\geq 1$ such ...
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27 views

If a self-adjoint operator has a mixed spectrum, is it necessarily bounded from below?

Let $A:\mathcal{D}(A)\subsetneq\mathscr{H} \rightarrow \mathscr{H}$ be an unbounded self-adjoint operator in a complex separable Hilbert space. Let $\sigma_{\text{pp}}(A)\neq \emptyset$ and $\sigma_{\...
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32 views

Is translation operator norm-preserving on $L^\infty(\mathbb{R})$?

Suppose $f \in L^\infty(\mathbb{R})$ and consider the translation operator $T_h: L^\infty(\mathbb{R}) \rightarrow L^\infty(\mathbb{R})$ given by $T_h(f) = f(x-h)$ for some $h \in \mathbb{R}$. I was ...
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1answer
27 views

Integrals of operator valued and Hilbert space valued functions

Consider a Hilbert space $H$ and the set of bounded operators $B(H)$. I am interested in integrating functions of the form $f:X \to H$ and $A: X \to B(H)$ where $X$ is generally a measure space, but ...
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23 views

Matrices equation $A^nX-XB^n=\sum_{k=0}^{n-1} A^{n-1-k}(AX-XB)B^k$, $X-A^nXB^n=\sum_{k=0}^{n-1}A^k(X-AXB)B^k$

It seems that for matrices $A,B$ formulae hold: $A^nX-XB^n=\sum_{k=0}^{n-1} A^{n-1-k}(AX-XB)B^k$, $X-A^nXB^n=\sum_{k=0}^{n-1}A^k(X-AXB)B^k$, but I'm having difficulties proving them by induction. ...
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15 views

Contractions semigroup

Let $H$ Hilbert, $\{e_k\}$ orthonormal basis, $A \colon D(A) \subset H \to H$ generator of a strongly continuous semigroup $e^{At}$ and $A$ such that $$Ae_k=-\lambda_k e_k$$ for some eigenvalues $\...
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1answer
18 views

On closed-graph theorem

Let $(X, \|\cdot\|_X)$, $(Y, \|\cdot\|_Y)$ be two Banach spaces and $T: X \to Y$ a linear operator. Then: $T$ is continuous $\iff$ $T$ is closed where "$T$ closed" means its graph $G(T)\...
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40 views

Norm of the integral operator [closed]

Let $T_K$ be an operator $$T_Kf(t) = \int_\Omega K(t,s)f(s) \, ds$$ where $f \in L^2(\Omega,m)$ and $K \colon\Omega \times \Omega \rightarrow \mathbb{C} $ where $m$ is a $\sigma$-finite measure. ...
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31 views

Resolvent of a operator with variable coefficients $x^2\partial_{x}^2$

I am studying the resolvent of operators and the following question has arisen: In $\mathbb{R}$, for the Laplacian operator $\partial_{x}^2$, it is natural that the resolvent operator is $R_{\lambda}=\...
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15 views

Essential spectrum of a product of self-adjoint operators which have positive essential spectrum

Given an infinite-dimensional Hilbert space $H$, a bounded linear self-adjoint operator $A:H\to H$ and a bounded linear invertible operator $B:H\to H$. If both $A$ and $B$ have positive essential ...
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22 views

Composition of operators and their derivatives

I'm a newbie in mathematical physics and is currently reading Sadri Hassani's book entitled Mathematical Physics - A Modern Introduction to Its Foundations. I came upon this page in the book and I ...
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1answer
59 views

Extension of functional calculus of continuous functions

On Reed & Simon's book, we can find the following theorem, which is called the continuous functional calculus. Notation: $\sigma(A)$ is the spectrum of the operator $A$, $\mathscr{L}(\mathscr{H})$ ...
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69 views

Numerical range of a symmetric positive matrix as an operator acting on Hilbert space

Let $A$ be a $3\times 3$ real constant symmetric positive matrix, $\Omega\subset\mathbb{R}^3$ a bounded Lipschitz domain and $(L^2(\Omega))^3$ the space of square integrable functions on $\Omega$. I ...
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19 views

Real determinant line bundle is trivial for complex linear operators

In Fredholm theory we can define the determinant line of a linear Fredholm operator $F$ to be $$\det F=\Lambda^\text{top}\operatorname{Ker}F\otimes(\Lambda^{top}\operatorname{Coker}F)^*$$ For a family ...
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Doubt in notation in Pseudodifferential and Singular Integral Operators: An Introduction with Applications, Abel

In Pseudodifferential and Singular Integral Operators: An Introduction with Applications by Abel, page 47 Query. How is the function $\left\{\mathcal{X}_{\epsilon}(y,\eta)\right\}_{0<\epsilon<1}...
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Perturbation of the spectrum

In a lecture note, in the demonstration of a result it is said that "using analytic perturbation theory" it is possible to deduce certain things. The reference is Kato's classic book, "...
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49 views

A $\ast$-representation of a $C^{*}$-algebra on a Hilbert space $H$ that is non-degenerate on a dense subspace $D\subset H$.

Suppose that $\pi$ is a $\ast$-representation of a $C^{\ast}$-algebra $A$ on some Hilbert space $H$. We say that $\pi$ is non-degenerate if for each $0\neq y\in H$ there is an $a\in A$ such that $\pi(...
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14 views

Essential spectrum of a constant matrix as an operator defined on a product of Hilbert spaces

Let $M$ be a $3\times 3$ constant real matrix, $\Omega\subset\mathbb{R}^3$ a bounded Lipschitz domain, and define $(L^2(\Omega))^3$ as the space of square integrable functions on $\Omega$. I want to ...
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1answer
29 views

About an inequality in the proof of maximal function

In the proof of the fact that if $f\in L^p$, where $1<p\leq L^p$, then $Mf\in L^p$ and $$||Mf||_p\leq A_p ||f||_p$$ ($Mf$ is maximal function) Stein says $$|f(x)|\leq |f_1(x)|+\alpha /2 (*)$$ and ...
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64 views

Why is the operator $P = - i \frac{d}{dx} $ not bounded in this domain?

the domain is: $$D(P) = \{f \in C^{\infty}(\mathbb{R}) \cap L^2(\mathbb{R})\}$$ I think that is because some functions that belong to $D(P)$ should not be bounded at infinity. The text I am studying ...
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1answer
22 views

For given Pontryagin space the norm induced topology is not depending on the fundamental decomposition

For given Hilbert space $H$ consider its corresponding antispace, i.e. the vector space endowed with its negative inner product. $\Pi$ is now called a Pontryagin space if it can be written as a direct ...
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41 views

The image of a Fredholm operator under an isomorphism : did the index still the same?

Let $X$, $Y$ be infinite Hilbert spaces, $T:X\to Y$ a bounded linear invertible operator and $A:X\to X$ a bounded linear operator. Consider then the following application: $$ \mathcal{F}: \mathscr{L}(...
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1answer
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Doubt in understanding the proof that a character of a unital abelian Banach algebra is of norm 1.

In the following theorem Theorem : Let $A$ be a unital abelian Banach algebra. If $\tau \in \Omega(A)$, then $\|\tau\|=1$. Proof. If $\tau \in \Omega(A)$ and $a \in A$, then $\bf \tau(a) \in \sigma(...
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Doubt in one step of the proof in order to show $u(X)$ is closed in $Y$, where $X$ and $Y$ are Banach spaces and $u$ is bounded

In the following theorem : Let $X, Y$ be Banach spaces and $u \in B(X, Y) .$ Suppose that there is a closed vector subspace $Z$ of $Y$ such that $u(X) \oplus Z=Y$. Then $u(X)$ is closed in $Y$. The ...
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1answer
88 views

Eigenvectors not in the Underlying Hilbert Space

Eigenvectors for the momentum operator: $P=-i\dfrac{d}{dx}$ are functions $f_p(x):=e^{ipx}$ which are not in $L^2(\mathbb{R})$. The notion of orthogonality can be made rigorous by use of Fourier ...
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1answer
31 views

Almost-Mathieu operator in the irrational case: show that the spectrum does not depends on the phase

Consider the Almost-Mathieu operator $H(\theta)$ defined for any $u \in \ell^2(\mathbb{Z})$ by $$ [H(\theta)\,u]_n = u_{n+1} + u_{n-1} + 2 \lambda \cos \big[2\pi (\alpha n + \theta)\big]\, u_n. $$ I ...
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Convergence of self-adjoint operator in norm resolvent sense

Let $A_n$ and $A$ be self-adjoint and $A_n\rightarrow A$ in norm resolvent sense i.e. $R_{A_n}(z)\rightarrow R_A(z)$ converges in norm for some $z\in\mathbb{C}/\mathbb{R}$. Also let $A$ to be bounded....
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70 views

Computation of the trace of a convolution operator

I try to understand a particular step on page two from the research paper "A multidimensional analog of a limit theorem of G. Szegö". https://iopscience.iop.org/article/10.1070/...
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23 views

Inverse Operator with a Matrix of operators as entries

Let $H$ a Hilbert space, and let $A:H\times H\rightarrow H\times H$ an operator given by $$A \begin{pmatrix}h_1 \\ h_2 \end{pmatrix}=\begin{pmatrix}P & Q \\ R &S \end{pmatrix}\begin{pmatrix}...
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1answer
30 views

Inserting a projection between (un)bounded operators

Consider a Hilbert space $\mathcal{H}$ and let $A$ be a bounded operator on $\mathcal{H}$, $B$ a possibly unbounded operator with domain $\mathcal{D}(B) \subset \mathcal{H}$ such that the composition $...
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1answer
32 views

Hardy-Littlewood maximal function is not in $L^1$.

It is from my text book. We know that the statement: if $f\in L^1(\mathbb{R}^n)$ then $Mf\in L^1(\mathbb{R}^n)$ is not true. In fact, if $Mf\in L^1(\mathbb{R}^n)$, then $f=0$. We can show this. If $a&...
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1answer
30 views

How is $\mathcal L(E)$ graded? Question about Hilbert $B$-module

I am reading G.G.KASPAROV's THE OPERATOR AT-FUNCTOR AND EXTENSIONS OF C*-ALGEBRAS Let $B$ be a C*-algebra and let $E$ be a $B$-right-module. Assume there is a $B$-valued inner product on $E$ which (...
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20 views

Graded algebra associated to finite groups [closed]

I would like to know many examples of graded algebra that arise or associated from a finite group. Please give me some reference. Thanks in advance
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29 views

Contraction proof of averaging operator: $\operatorname{Var}_\nu (M f) \leq (1 - \kappa)^2 \operatorname{Var}_\nu (f)$

Let $(X, d, m)$ be an ergodic random walk on a metric space, with invariant distribution $\nu$. Suppose that the coarse Ricci curvature of $X$ is at least $\kappa > 0$ and that the average ...
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36 views

Is the inverse Laplacian bounded in $\mathbb{R}^{2}$?

I'm searching for an inequality in the form $$\forall s>2,\quad\forall u\in H^{s-2}(\mathbb{R}^{2}),\quad\|\Delta^{-1}u\|_{H^{s}(\mathbb{R}^{2})}\lesssim\|u\|_{H^{s-2}(\mathbb{R}^{2})}$$ where $H^s$...
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3answers
62 views

How can we show $\lim_{\lambda\to\infty}\left\|\lambda R_\lambda(A)x-x\right\|_E=0$?

Let $A$ be a (possibly unbounded) linear operator on a Banach space $E$ such that $(0,\infty)\subseteq\rho(A)$ and $$\left\|R_\lambda(A)\right\|_{\mathfrak L(E)}\le\frac1\lambda\;\;\;\text{for all }\...
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1answer
47 views

What is the value of knowing the eigenfunctions of an operator?

Let $f_i$ be the set of functions of operator $O$ such that $Of_i = \lambda_if_i\;\;\lambda_i \in \mathbb{R}$ In linear algebra, there are a number of uses of the eigenvalues and eigenvectors. Are ...
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20 views

$1/2$ Hölder estimate on the difference of Hilbert--Schmidt operators

Let $A,B$ be trace class, self-adjoint, and positive operators on some separable Hilbert space $H$. Then, does an inequality of the following form hold true: $$ \lVert A^{1/2}-B^{1/2}\rVert_{\mathrm{...
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1answer
24 views

Dimension of the subset of linear operators.

Let $v$ be a fixed non-zero vector of an $n$-dimensional real vector space $V$. Let $P(v)$ be the subspace of the vector space of linear operators on $V$ consisting of those operators that admit $v$ ...

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