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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

Show that an integral operator from $H^{1/2}(\partial D)$ to $L^2(\mathbb{R}^{3-}\setminus\partial D)$ is bounded

Let $\partial D\subset\mathbb{R}^{3-}$ be Lipschitz regular. I would like to show that the integral operator $$L(g)=\int_{\partial D}\frac{1}{4\pi}\left(-\frac{(x-y)(x-y)^T}{|x-y|^5}+\frac{1}{|x-y|^...
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1answer
19 views

Do the transpose operator and the adjoint operator have same ranges and kernels?

Let $A$ be a bounded operator on a comlex Hilbert space $H$. Do we have : $R(A^t)=R(A^*)$ And $Ker(A^t)=Ker(A^*)$ ? Thank you!
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1answer
35 views

Is it true that $U^{-1}\overline{A}U=\overline{U^{-1}AU}?$

Let $X_0$ and $X_1$ be Banach spaces and suppose that $A:D(A)\subseteq X_0\times X_1$ is a closable linear operator. If $U: X_0\to X_1$ is a unitary operator, is it always true that $$U^{-1}\overline{...
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2answers
28 views

Normal plus compact is Fredholm with index 0

Let $T, K \in B(H)$, $TT^{*} = T^{*}T$, $K$ compact. Why is $T+K$ Fredholm and $ind(T +K) = 0$? Is it even true? I have forgotten all I knew about Fredholm theory and I need that result for proving ...
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1answer
21 views

Comparison of projections in $B(H)$

Suppose $P$, $Q$ are two non-trivial projections in $B(H)$, can we deduce that $P\leq Q$ or $Q\leq P$?
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1answer
42 views

Is the point spectrum always countable?

I have this very simple question. Premise: Let $A$ be a linear densely defined symmetric/self-adjoint operator in a complex separable Hilbert space $\mathcal H$ (typical example in Quantum Mechanics)....
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1answer
34 views

compute the sprectrum of the sum of orthogonal projections [closed]

Suppose $\{p_i\},i=1,\cdots,n$ are different projections and they are mutually orthogonal in a $C^*$-algebra,how to compute the spectrum $\sigma(k_1p_1+\cdots+k_np_n)$ of $ k_1p_1+\cdots+k_np_n$, ...
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1answer
23 views

Corollay of Theorem 4.1.3 from Arveson's Book

from book A Short Course on Spectral Theory , William Arveson Corollary of Theorem 4.1.3 says Every Toeplitz operator $T_\phi,\phi \in L^\infty$, satisfies $\displaystyle\inf\{\|T_{\phi}+K\| \...
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27 views

The distribution of Projection, Uniformly Ergodic and Mean Ergodic operators in $\mathbb{B}(\mathbb{E})$

Let $\mathbb{E}$ be a (real or complex) Banach space. Let $\mathbb{B}(\mathbb{E})$ be the Banach Space of all continuous (bounded) linear operators from $\mathbb{E}$ into $\mathbb{E}$, with the ...
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1answer
44 views

Using functional calculus to show that $exp(S)exp(T)=exp(T)exp(S)$

Suppose $S$ and $T$ are two commuting normal operators in $B(H)$,how to use the continuous functional calculus to show that following conclusion? $exp(S)exp(T)=exp(T)exp(S)=exp(S+T)$? Can we derive ...
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1answer
60 views

A bounded, self-adjoint, positive operator $T$ induces a positive semidefinite quadratic form $\langle Tx,x\rangle$.

Let $H$ be a Hilbert space and $T\in \mathcal B(H)$ be a bounded, self-adjoint linear operator that is positive in the sense that $\sigma(T) \subset [0,\infty)$. Is there an elementary method of ...
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16 views

Exponential operators with derivatives

How one could possible calculate $\exp(-it\exp(iax)[d^2/dx^2 + c])\exp(ibx)$ or equivalently solve $i\frac{d}{dt}f(t,x) = \exp(iax)[d^2/dx^2 + c]f(x,t)$ with initial condition $f(0,x) = \exp(ibx)$? ...
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1answer
28 views

application of Borel functional calculus

Suppose for any postive operator $T \in B(H) $,we have a real bounded Borel function $f_{T}$ on $\sigma(T)$. If we have a non-trivial projection $P$on a Hilbert space $H$ and $k>0$.Let $P^{\perp}=...
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18 views

definition of anti-unitary operator

The definition of anti-unitary operator is given as following: A bounded anti-linear operator $U$ is anti-unitary if $UU^*=U^*U=1$ But I found another definition in Wiki:an anti-unitary operator $U$...
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1answer
20 views

difference between continuous functional calculus and borel functional calculus

When $N$ is a normal operator on $H$ with spectral measure $E$,let $B(\sigma(N))$ be the $C^*$ algebra of bounded Borel functions on $\sigma(N)$ ,we have the map $\psi\mapsto \psi(N)$,which is a ...
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1answer
21 views

continuous functional calculus about exponential function and log function

Suppose $S$ and $T$ are self-adjoint operators in $B(H)$. we can use the following continuous functional calculus : $C(\sigma(S)) \to C^*(S)$, $f\mapsto f(S)$ $C(\sigma(T)) \to C^*(T)$, $g\mapsto g(T)...
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28 views

Polar decomposition for general $C^*$-algebra

Suppose $A$ is a $C^*$-algebra,for each element $a \in A$ and $r \in (0,1)$,prove that there exists $u \in A$ such that $a=u(\sqrt{a^*a})^r$ I can prove it for normal elements. But how about the ...
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0answers
37 views

Minimize $\langle(1-\kappa)^{-1}f,f\rangle$ for a parameter-dependent integral operator $\kappa$

I've got a contractive self-adjoint linear integral operator $\kappa$ of the form $$(\kappa g)(x):=g(x)+\int\lambda({\rm d}y)k(x,y)(g(y)-g(x))\;\;\;\text{for }g\in L^2(\mu),$$ where $k$ depends on the ...
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1answer
23 views

continuous functional calculus about two commutative opertaors

Suppose $S,T$ are two commuting normal operators in $B(H)$,how to use the continuous functional calculus to show that $f(S)g(T)=g(T)f(S)$,where $f\in C(\sigma(S)),g\in C(\sigma(T))$? There is an ...
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0answers
36 views

Point spectrum empty and spectrum {0,1} [closed]

I need example of a bounded linear operator $$A: l^2 \rightarrow l^2 $$ such that point spectrum of A is empty and spectrum of A is {0,1}.
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1answer
38 views

positive invertible elements in some $C^*$ algebra

Suppose we take $A=B(H)$,where $H$ is a complex Hilbert space.Let $S,T$ be two positive invertible elements in $A$,$K=\{m>0,S\leq m T\}$, when $dim(H)=1$,it is easy to see that $K$ has a minimum,...
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1answer
34 views

BCH formula for finding $\hat{C}$ in equation $e^{\hat{A}}e^{\hat{B}}=e^{\hat{A}+\hat{B}}e^{\hat{C}}$

Asume we have 2 noncommutative operators $\hat{A}, \hat{B}$ but they comply this condition $$ [\hat{A},\, [\hat{A},\, \hat{B}]]=[\hat{B},\, [\hat{A},\, \hat{B}]]=0. $$ Find what equals to operator $\...
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0answers
36 views

Boundedness of Hilbert-Hankel operator

This is an exercise in Lax. The Hilbert-Hankel operator is defined to be $f\rightarrow g(r)=\int^\infty_0 \frac{f(t)}{t+r}dt$. The question is to show the operator is a bounded map of $L^p(\mathbb{R^+...
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18 views

Is the image by a closed range operator of a closed subspace closed too?

Let $H$ be a Hilbert space and suppose we have a bounded closed range operator $T\in B(H)$. Is $T(M)$ closed for every closed subspace $M\subset H$ ? Thank you !
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20 views

Representation formula for the $n$th power of a self-adjoint operator in terms of the spectral measure

Let $H$ be a $\mathbb R$-Hilbertspace, $A$ be a self-adjoint linear contraction on $H$ and $E:\mathcal B(\mathbb R)\to\mathfrak L(H)$ denote the spectral measure associated with $A$. By contractivity, ...
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2answers
54 views

homomorphisms of B(H)

Let $\varphi\colon B(H)\to B(H)$ is an injective $\ast-$homomorphism ($\varphi$-linear, $\varphi(xy)=\varphi(x)\varphi(y)$, $\varphi(x^{\ast})=\varphi(x)^{\ast}$), where $H-$separable Hilbert space. ...
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1answer
15 views

Orthonormal basis for Hilbert Schmidt operators

We know that $B_2(H)$ (collection of Hilbert Schmidt operators on $H$) forms a Hilbert space with respect to the inner product $\left<A,B\right>=\text{tr}(B^*A)$. I was wondering if there is ...
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1answer
91 views

Use of the symbolic operator $\left[e^{\frac{\partial}{\partial x}}\right]$ in Taylor's expansion

We know that Taylor's series expansion of a generic function $f(x)$, in the abscissa point $x_0\in\mathbb{R}$ is given by: $$f(x)=\sum_{n=0}^{\infty}\frac{(x-x_0)^n}{n!}f^{(n)}(x_0) \tag{1}$$ $$f(x)=...
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1answer
46 views

Spectrum of complex normed space bounded operator

This is a problem from Functional Analysis by B.V Limaye which I am finding difficult to deal with: Let $X$ be a normed space over complex field and let $$ A: X\rightarrow X $$ be a bounded linear ...
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23 views

spectrum of a unital algebra

Suppose $A$ is a unital Banach algebra, $a\in A$ we can define the spectrum of $a$ as following: $\operatorname{sp}(a)=\{\lambda\in \Bbb F:\lambda \cdot 1_{A}-a$ is not invertible $\}$. My question ...
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28 views

Differentiability of spectral functions of self-adjoint operators on Hilbert spaces

Is there a generalization of the differentiability result of spectral functions of Hermitian matrices (see Theorem 1.1 here: https://pdfs.semanticscholar.org/b284/3d6c16f54fd5070e9d9e0c19c14a9f43ed4f....
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2answers
26 views

Is the function mapping a self-adjoint operator on a Hilbert space to the maximum of its spectrum differentiable?

Let $H$ be a $\mathbb R$-Hilbert space. Is $$\left\{A\in\mathfrak L(H):A\text{ is self-adjoint}\right\}\to\mathbb R\;,\;\;\;A\mapsto\max\sigma(A)\tag1$$ differentiable? We know that this is true in ...
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17 views

If $A$ is a multiplication perturbed by an integral operator on $L^2$, what can we say about the maximizer of $\sup_{\|f\|=1}⟨(1-A)^{-1}f,f⟩$?

Let $(E,\mathcal E,\mu)$ be a probability space and $$L^2_0(\mu):=\left\{f\in L^2(\mu):\int f\:{\rm d}\mu=0\right\}$$ $r:E\to[0,1]$ be $\mathcal E$-measurable and $$M_rf:=rf\;\;\;\text{for }f\in L^2(\...
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1answer
39 views

Semigroups: given $\|S_T f\| \leq C$, what can I say about $\|S_t f\|$ for $t \leq T$?

Assume that we have a strongly continuous semigroup $(S_t)_{t \geq 0}$ of linear operator on a Banach $X$ such that for all $f \in X$, $$\|S_T f\| \leq C \|f\|, $$ for some constant $C > 1$ and ...
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24 views

How does $\sigma(T^*T)$ depends on $\sigma(T)$?

Is there any relation between $\sigma(T^*T)$ and $\sigma(T)$ ? Thank you!
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1answer
29 views

Every weak*-closed set is the range of an adjoint operator

Let $X$ be a normed linear space and let $N$ be a weak*-closed subspace of $X^*$. How to show that there exists a normed linear space $Y$ and $T\in B(X,Y)$ such that $T^*(Y^*)=N$? I feel that if $T$ ...
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38 views

for linear Operators is it true that $(A^*)^{-1}=I$, whrere $I$ is the neutral element?

If A is a linear operator between Hilbert spaces. Is the following statement true? $(A^*)^{-1}A=I$, where $I$ is the neutral element so that $A^{-1}A=I$
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2answers
70 views

In which kinds of books can I find the definition of exponential operator?

I have been studying real and functional analysis recently, but I hadn't encountered an exponential operator such as $e^{L}$ until I recently read a paper. In which kinds of books can I find the ...
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1answer
22 views

Let $T:X\longrightarrow Y$ be a weakly compact operator between Banach spaces. Prove range of $T$ is closed iff range of $T$ is reflexive

The proof is from Robert Megginson, Introduction to Banach Spaces (3.5.6) However I dont understand why $B_{T(X)}$ is weakly compact, I've been using Eberlein - Smulian Theorem to prove the ...
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0answers
25 views

Show that this integral operator is Hilbert-Schmidt

Let $(E,\mathcal E)$ be a measurable space, $\mu$ and $\nu$ be $\sigma$-finite measures on $(E,\mathcal E)$, $q\in L^2(\mu\otimes\nu)$ and $$Qg:=\int\nu({\rm d}y)q(\;\cdot\;,y)g(y)\;\;\;\text{for }g\...
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1answer
39 views

Can we show that $\sup_{\left\|x\right\|_H=1}\langle Ax,x\rangle_H$ is attained at the supremum of $\sigma(A+A^\ast)$?

Let $H$ be a $\mathbb R$-Hilbert space and $A\in\mathfrak L(H)$. Consider the following optimization problem: $$\sup_{\left\|x\right\|_H=1}\langle Ax,x\rangle_H.\tag1$$ We may note that $A+A^\ast$ is ...
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0answers
33 views

Spectral measure for symmetric matrices

Let $H$ be a $\mathbb R$-Hilbert space and $A$ be a bounded linear self-adjoint operator on $H$. Let $\mathcal M_b(\sigma(A))$ denote the $\mathbb R$-vector space of bounded Borel measurable functions ...
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1answer
38 views

Generalization of a symmetric eigenvalue problem for linear operators on a Hilbert space

Let $A\in\mathbb R^{n\times n}$. Which assumptions on $A$ do we need and how can we show that $$\max_{\substack{z\in\mathbb R^n\\|z|=1}}\langle Az,z\rangle\tag1$$ is attained at the unit eigenvector $...
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1answer
35 views

An easy application of the spectral theorem for a self-adjoint operator on $L^2$

Let $(E,\mathcal E,\mu)$ be a probability space, $$L^2_0(\mu):=\left\{f\in L^2(\mu):\int f\:{\rm d}\mu=0\right\}$$ and $$U:L^2(\mu)\to L^2(\mu)\;,\;\;\;f\mapsto\int f\:{\rm d}\mu=\langle 1,f\rangle_{L^...
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0answers
38 views

SG pseudodifferential operators, Fredholm implies elliptic

How can i prove that an operator $T_\sigma$ with $\sigma\in S^{m_1,m_2}$ ; $(m_1,m_2)\in(-\infty,\infty) $ $T_\sigma: H^{s_1,s_2,p}\rightarrow H^{s_1-m_1,s_2-m_2,p}$ which is Fredholm is elliptic? I ...
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0answers
74 views

How to show this vector identity involving Laplacian?

Define an operator $L_{+}$ as follows: $$L_{+} = -\Delta + 1 - pQ^{p-1}$$ Let $Q$ be the solution to the nonlinear PDE: $$Q-\Delta Q - |Q|^{p-1}Q =0.$$ Let $$Q_1 = \left(\frac{2}{p-1} + x\cdot \Delta\...
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1answer
58 views

Spectral decomposition of the operator $L^2\to L^2,f\mapsto\int f$

Let $(E,\mathcal E,\mu)$ be a probability space, $$L^2_0(\mu):=\left\{f\in L^2(\mu):\int f\:{\rm d}\mu\right\}$$ and $$U:L^2(\mu)\to L^2(\mu)\;,\;\;\;f\mapsto\int f\:{\rm d}\mu=\langle 1,f\rangle_{L^2(...
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1answer
35 views

What is the square-root of the nonnegative self-adjoint operator $L^2\to L^2,f\mapsto\int f$?

Let $(E,\mathcal E,\mu)$ be a probability space. Since $\mathbb R$ is naturally embedded into $L^2(\mu)$, we may consider the operator $$U:L^2(\mu)\to L^2(\mu)\;,\;\;\;f\mapsto\int f\:{\rm d}\mu.$$ ...
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0answers
52 views

Can we generalize the following result on the spectral gap of a reversible Markov transition matrix?

Let $(E,\mathcal E,\pi)$ be a probability space, $$L^2_0(\mu):=\left\{f\in L^2(\mu):\mu f=0\right\}$$ and $\kappa$ be a Markov kernel on $(E,\mathcal E)$ such that $\mu$ is reversible with respect to $...
2
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0answers
22 views

The spectrum of an irreducible reversible Markov kernel is contained in $[-1,1)$

Let $(E,\mathcal E)$ be a measurable space, $\kappa$ be a Markov kernel on $(E,\mathcal E)$, $\mu$ be a probability measure on $(E,\mathcal E)$ reversible with respect to $\kappa$ and $$L^2_0(\mu):=\...