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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Spectrum of a positive operator

We know that if $A$ is a self-adjoint unbounded operator on a Hilbert space $(H;\left<.,.\right>)$ then $\sigma(A) \subset \mathbb R$. Now, how it can be shown that if $A$ is more positive i.e. ...
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Properness of isometric actions of discrete groups on affine Hilbert spaces

I've been reading Valette's introduction to the Baum-Connes conjecture and as I read the example of a construction of a (model of the) classifying space for proper actions of $\Gamma$ (discrete) given ...
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34 views

When can I Taylor expand a function of an operator?

1-) Is the expression $f(A) = \sum_n \frac{f'(0)}{n!}(A)^n$ always meaningful for any diagonalizable linear operator $A$ and for any analytic function $f$? This seems strange to me because then I ...
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38 views

A property of the Volterra operator

I was reading a paper and I came across the Volterra operator $$(Vf)(x)=\int ^x _0 f(t) dt$$ And its adjoint $$(V^*f)(x)=\int^1 _x f(t) dt$$ It also says that a simple and useful identity is ...
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Strong Convergence of Fredholm Operators, as used in Callias' proof of his index theorem

In his paper Axial Anomalies and Index Theorems on Open Spaces, Callias provides a wonderful index theorem $$\mathrm{index}(L)=\lim_{z\to0} \mathrm{Tr}B_z\quad\text{where} \quad B_z=\frac{z}{L^\dagger ...
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Is the set of adjoint operators weak* closed?

Suppose we have a Banach space $X$ and a net of bounded operators $(T_\gamma)$ on $X$ such that $T_\gamma^*\to S$, for some bounded operator $S$ on $X^*$, where the convergence is with respect to the ...
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30 views

Understanding Operator Norm of Matrices

Let $X$ denote the vector space of $n\times n$ complex matrices. To every matrix $A\in X$ one can associate two operator norms: Thinking of $A$ as a map $A\colon \mathbb{C}^n\to \mathbb{C}^n$ or ...
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40 views

Spectrum of an Operator on a Banachspace

Claim: Let $A$ be a bounded linear operator on a Banachspace $\mathfrak{X}$. Denote $\sigma(A)$ as the spectrum of A. Let $\lambda$ be a point in the boundary of the $\sigma(A)$. Then there exist a ...
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34 views

Why is it true that the multiplication operator in a reproducing kernel Hilbert space is always continuous?

In my functional analysis I was met with this seemingly trivial theorem on RKHS If $ \mathbb{H} $ is a reproducing Kernel Hilbert Space and we have a multiplier $ \phi $ meaning it satisfies $ ...
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A conjecture about traces of projections

Let $M_n$ denote the space of all $n\times n$ complex matrices. Define $\tau:M_n\rightarrow \mathbb{C}$ by $$\tau(X)=\frac{1}{n}\sum_{i=1}^n x_{ii},$$ where of course $X=[x_{ij}]\in M_n$. Recall that ...
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Construction of Sobolev space

I am reading about the construction of Sobolev spaces from $L^2$. In the book I read (Introduction to Partial Differential Equations, by Gerald B. Folland) that the operator used to construct those ...
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Is adjoint operator of generator of an analytic semigroup be a generator of analytic semi-group?

Let $X$ be a Banach space. The adjoint semigroup $\{T(t)^\prime:t\ge 0\}$ consisting of all adjoint operators $T(t)^\prime$ on the dual space $X^\prime$ is, in general, not strongly continuous where ...
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15 views

Are there identities which will make calculating $\Delta \frac{e^{ik|x|}}{|x|}$ more efficient and clearer?

Let $x = (x_1, x_2, x_3)$ be a point in $\mathbb{R}^3$. I am trying to calculate $$\Delta \frac{e^{ik|x|}}{|x|}$$ Doing this 'manually' be calculating second derivatives is really long and tedious ...
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For $\{T_n\}$ and $T$ positive and self-adjoint, show $T_n \stackrel{SR}{\to} T$ iff $(T_n + I)^{-1} \stackrel {s}{\to} (T + I)^{-1}$

For $\{T_n\}$ and $T$ positive and self-adjoint, show $T_n \stackrel{SR}{\to} T$ (i.e. $T_n \to T$ in the strong resolvent sense) iff $(T_n + I)^{-1} \stackrel {s}{\to} (T + I)^{-1}$ (i.e. $(T_n + ...
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35 views

What are useful mappings (operators) in image reconstruction

I'd like to ask the technician mates to provide some information regarding mappings and image reconstruction operators. Please, if possible, provide some articles and helpful discussions about useful ...
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37 views

Proof that a linear operator is continuous.

Could somebody please verify the following proof I have attempted? It seems too simple so I am worried I have done something wrong.. Many thanks Let $T:(X,\|.\|_X)\to (Y,\|.\|_Y)$ be a linear map ...
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21 views

Interchanging Limit and Integral sign

I'm reading a book on composition operators, and the author makes the following claim: Given a self-map of the unit disc, and a $H^2$ function $f$, where $H^2$ is the Hardy space, if we fix a radius ...
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20 views

If a contraction and its adjoint converge to zero both does that mean the contraction satisfies $ ||Th|| < h $

I just met this in my functional analysis on contractions which got me stumped: Let $ \mathbb{H} $ be a Hilbert space and let T be a contraction operator on $ \mathbb{H} $ (meaning $ ||T|| \leq ...
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33 views

Does a contraction converging in power series necessarily lead to the operator being a proper contraction?

I was recently met with this in my functional analysis class on which I am stuck: Let $ \mathbb{H} $ be a Hilbert space and let T be a contraction operator on $ \mathbb{H} $ (meaning $ ||T|| ...
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41 views

Two sequences of operators on Hilbert Space

Let $H$ is some Hilbert Space, and $a_n,b_n \in B(H)$ is sequences of some operators on it. We know, that $a_n b_n$ converges to $v$ by norm. We also know, that all $b_n a_n$ are strictly positive. ...
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A naïvely constructed extrapolation of a self-adjoint operator. Is it self-adjoint?

Let $\mathcal{H}$ be a real Hilbert space and let $A\colon D(A)\subset \mathcal{H}\to \mathcal{H}$ be an unbounded operator. Consider also a Hilbert triple $$ \mathcal{H}_+\subset \mathcal{H}\subset ...
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Index of a derivative operator on a circle

Let $D: C^{1}(S^{1}) \rightarrow C(S^{1})$ be an operator defined as $D(f)=f'$. I would like to find its index (on the road proving that it's a Fredholm operator). First, if $f \in ker(D)$, then ...
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20 views

Formula connecting the resolvent opeartor andthe spectral density?

I want to know if it is a formula connecting the resolvent opeartor $(\lambda - T)^{-1}$ for a selft-adjoint operator $T$ and its spectral density $e_{\lambda}$. Thank you in advance
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Formula connecting the resolvent and the heat kernels

Using the well known formula connecting the resolvent and the heat operators associated to a selft-adjoint opeartor $A$ \begin{align} (\zeta - A)^{-1} = \int_{0}^{\infty} e^{-\xi t} \, e^{t A} dt; ...
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1answer
15 views

Index of a differential operator

Let's consider an operator $D: C^{m+n}[a, b] \rightarrow C^{m}[a, b]$, defined as $D(y(t)) = y^{(n)}+a_{n-1}y^{(n-1)}+\ldots+a_{1}y'+a_{0}$, $a_{k} \in C^{m}[a, b]$. I would like to prove that it is ...
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30 views

Linear operators and weak star topology.

Let $\mathbb{L} = \operatorname{span}\{\delta_t : t \in [0, 1]\} $, where $\delta_t \in C[0,1]^*, \delta_t(f) = f(t).$ How to prove that the linear functional $G(f) = \int_0^1 f(t) dt$ belongs to ...
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$L_p$ version of Toeplitz extension

There is the well-known extension of $C^*$-algebras $0\rightarrow K\rightarrow\mathcal{T}\rightarrow C(S^1)\rightarrow 0$ where $\mathcal{T}$ is the Toeplitz algebra (generated by the unilateral shift ...
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Finding the bound of a linear functional defined on $C[-1,1]$

Define the linear functional: $$f(x)=\int_{-1}^{0}x(t)dt-\int_0^1x(t)dt$$ On the normed space $C[-1,1]$ which consists of all contiuous functions on the interval. The norm is defined as: $\|x\|= ...
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Direct Integral: Scalars

Given a Borel space $\Omega$. Regard the Hilbert Space: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}_+:\quad\mathcal{H}:=\mathcal{L}^2(\Omega;\mu)$$ Denote the Borel Projections: ...
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265 views

Spectral Measures: Stone's Formula

Hilbert Space: $\mathcal{H}$ Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Spectral Measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad ...
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Does proper contraction on Hilbert space necessarily lead to convergence in norm to zero?

I was asked this in functional analysis class: Let $ \mathbb{H} $ be a Hilbert space and we are given an operator T satisfying: $ || Th || < ||h|| $ for all $ h \in H $. We are asked if ...
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1answer
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Bounded Operators, Unitary group

It's clear to me that if H is a self-adjoint bounded operator on a Hilbert space, then the bounded operators $$U_t :=\sum_{n=0}^\infty (iHt)^n / n!$$ are unitary for all $t\in \mathbb{R}$. How do I ...
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Show the operator $T$ is bounded where $T: l^2 \to c$

Let $c$ be the Banach space of all sequences $y=(n_j)^\infty _1 $ such that $\lim_{j \to \infty} n_j$ exists. The norm on $c$ is given by $||y||=\sup_j |n_j|$. Consider the operator $T: l^2 \to c$ ...
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Decomposition an operator in terms of symmetric and anti-symmetric components

In linear algebra, we can write any operator as the sum of a symmetric and skew-symmetric parts: $$A=A^{\mathrm{sym}}+A^{\mathrm{skew}}$$ where $$A^{\mathrm{skew}}=\frac{1}{2}(A-A^T)$$ and ...
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19 views

Does strict contraction lead to convergence to zero in norm?

In my functional analysis class I was asked this question which got me stuck: Let $ \mathbb{H} $ be a Hilbert space and we are given an operator T satisfying: $ || T || < 1 $ in the ...
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Index of an element in C*-algebra

Suppose that $x$ is an element of abstract $C^*$-algebra $A$. For example if $x$ is normal, i.e. $x^*x=xx^*$ then if we use any representation $\pi$ of $A$ on some Hilbert space $H$ then $\pi(x)$ will ...
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Uniqueness of element in infinite dimensional Hilbert space

Suppose $H$ is an infinite Hilbert space where $\{e_k:k\in \mathbb{Z}\}$ is a total orthonormal family. Let $H_1=\overline{span{(e_k: k=0, 1,2,\cdots})}$ and $H_2=\overline{span{(e_{-k}+ke_k: ...
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52 views

Equivalence and rank equivalence

Let $A$ be a $*$-algebra. Let $P(A)=\bigcup_{n=1}^\infty\{p\in M_n(A):\text {$p$ is a projection} \}$. By projection I mean $p=p^*=p^2$. Define the an equivalence relation on $P(A)$ by $p \sim q ...
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What are the projections of a commutative C* algebra?

I am aware that the commutative C* algebra is $C_0(X)$ for some nice space $X$ but I cannot figure out what the projections should be. The natural candidates (indicator functions on nice subsets of ...
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Under what conditions are such operators well defined?

Let H be a hilbert space, and $\phi_k$ a basis, one can define a "diagonal" operator $A$ by $A\phi_k=b_k\phi_k$, Is there a simple condition on the coefficients $b_k$ such that the operator is well ...
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$P+Q-PQ$ is a projection if and only if $PQ=QP$.

Let $\mathcal H$ is a Hilbert space and $P,Q:\mathcal H \to \mathcal H$ are projections. I want to show that $P+Q-PQ$ is a projection if and only if $PQ=QP$. If $PQ=QP$ clearly $P+Q-PQ$ is a ...
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37 views

How to check if an operator is invertible?

Let $T_1 : C[a,b] \to C[a,b]$ be an operator defined by $$T_ v(x)=\int^b _a (x-t)v(t) dt$$ where $a \leq x \leq b$ and $v \in C[a.b]$ How can you check if the operator $T_1$ is invertible or not?
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Is there an example of a non von Neumann algebra with this property?

What is an example of a $C^{*}$ subalgebra $A$ of $B(H)$ such that $A$ contains the identity $I_{H}$ and satisfies the following properties: 1) For every $T\in A$, The orthogonal projection ...
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In what sense are compact operators limits of finite-rank operators?

The convergence is in respect to what topology ? Can someone please write it mathematically ?
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How to calculate the norm of this operator?

Let $H$ be a separable Hilbert space and $(\phi_k)$ be a basis $A(t)$ is defined such as $A\phi_k=\exp(-t/k)\phi_k$. I am specifically interrested whether $\|A(t)\| \to 0$ when $t \to \infty$ or not, ...
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Compact operators are orthogonally equivalent to a diagonal matrix?

On Brezis's Functional Analysis, the last question of Problem 44 (near the end of the book) reads (modified to include context) Assume that the Hilbert space $H$ is separable and $T\in\mathcal ...
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$T$ is self-adjoint $\Rightarrow \exists$ positive $A,B$ such that $T=A-B$ and $AB=0$

I have a trouble by the following problem and I dont have any idea to solve it. can anybody give me a hint? Thanx in advance. Let $\mathcal H$ be a Hilbert space and $T:\mathcal H \to \mathcal ...
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Hadamard product involving operators

If we have two matrices $A=(a_{i,j})_{i,j}$, $B=(b_{i,j})_{i,j}$ representing linear and continuous operators from $\ell^2$ to $\ell^2$, it is known that the Hadamard product of them, $A\ast ...
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Spectral Measures: Poisson

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}H\to\mathcal{H}:\quad H=H^*$$ And its spectral measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad ...
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106 views

How does the general spectral theorem generalize the simpler versions?

I read the following version of the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I'm trying to understand why this is a generalization of the following version, ...