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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48 views

When is this matrix positive semidefinite?

Let us fix dimension $n$. Consider the $n \times n$ matrix \begin{equation} S_n=\begin{bmatrix} 1 & z & z & \cdots & z \\ \bar{z} & 1 & z & \cdots & z\\ \bar{z} & ...
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2answers
17 views

Domain of the closed extension operator is not the entire space?

Given a Banach space $X$, and a densely define linear operator $A:D(A)\subset X \rightarrow X$, we define the graph $$G_A= \{ (x,Ax) | x\in D(A)\}$$ which is a linear subspace of Banach space ...
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36 views

Self adjoint operators on Hilbert spaces are bounded

I think I have a proof that if $A: H\rightarrow H$ is a self adjoint operator on a Hilbert space $H$, then $A$ is bounded: We can use the closed graph theorem. Let $x_n \rightarrow x$ and $Ax_n ...
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17 views

the adjoint of left translation semigroup on L1(R) [closed]

Show that on the Lp-spaces, the right translation semigroups are the adjoints of the left translation semigroups; i.e., Tl(t) = Tr(t) for t ≥0.
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29 views

Eigenfunction of a selft-adjoint operator?

Let $A = \int_{0}^{\infty} \lambda dE(\lambda)$ be the spectral decomposition of a selft-adjoint operator $A$ on a Hilbert space $H$. Then the restriction operator $P_{\lambda}$ for $A$ is defined by ...
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16 views

Reference request: monotone and strongly monotone with respect to derivatives

Recall, let $H$ be a real Hilbert space. A mapping $F:H \rightarrow H$ is said to be monotone if $$ \langle F(u)-F(v), u-v\rangle\geq 0, \quad \forall u,v\in H; $$ strongly monotone if there exists ...
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31 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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1answer
31 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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11 views

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 ...
2
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1answer
28 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 ...
2
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1answer
30 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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51 views

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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2answers
41 views

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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1answer
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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1answer
42 views

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. ...
2
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1answer
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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18 views

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 ...
4
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1answer
36 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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0answers
19 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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1answer
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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31 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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1answer
19 views

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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30 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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34 views

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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1answer
14 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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0answers
18 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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18 views

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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6 views

$L_p$ 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 ...
3
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1answer
51 views

About a relation between isometries

If we have $(T_i)_{i=1}^N$, operators on a Hilbert space, that are also isometries and satisfy the following relation: $$\sum_{i=1}^NT_iT_i^*=Id\quad (1)$$ How can you prove that they must also ...
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1answer
26 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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14 views

Why we can write the spectral density$ e_{\lambda}$ in the following forms?

Let $e_{\lambda}$ be a the spectral density of i.e. $$ e_{\lambda} = \frac{dE_{\lambda}}{d\lambda} $$ associated to a the spectral function $E_{\lambda}$ for a self-adjoint operator $A$ on a complex ...
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3answers
39 views

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\|= ...
2
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1answer
28 views

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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0answers
21 views

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 $T: l^2 \to c$ ...
5
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1answer
61 views

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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0answers
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 ...
0
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1answer
16 views

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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0answers
29 views

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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1answer
23 views

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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1answer
18 views

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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1answer
30 views

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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2answers
66 views

$P+Q-PQ$ 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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0answers
16 views

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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24 views

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, ...
2
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1answer
50 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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2answers
35 views

$T$ 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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0answers
44 views

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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29 views

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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1answer
28 views

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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0answers
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 ...