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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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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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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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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 $...
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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):=\...
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1answer
23 views

Norm of the solution of a Poisson equation

Let $\kappa$ be a linear contraction (operator norm at most $1$) on a $\mathbb R$-Hilbert space $H$, $L:=1-\kappa$ and $f,g,\tilde g\in H$ with $$Lg=f=L\tilde g\tag1.$$ Are we able to show that $\left\...
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1answer
43 views

Proving that composition is continuous in the strong operator topology when the domain is bounded

Let $X$, $U$, $W$ be banach spaces and $L(X,U)$ be the set of bounded linear maps from $X$ to $U$ with the operator norm. Then The composition of linear maps is a mapping from $L(X,U) \times L(U,W)$ ...
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31 views

Coinciding Weak Operator and Strong Operator Topologies

Let $A \subset \mathcal{B}(H)$ a subalgebra, not necessarily a $*$-algebra. In Murphy's book 'C*-algebras and Operator Theory', in Remark 4.2.1 you can find a proof of the failure of strong ...
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36 views

A question on projection [closed]

Let $\mathcal H$ be a Hilbert space. Suppose $p,q$ are projections in $B(\mathcal H)$ such that $pqp=p.$ Is it true that $p\leq q$?
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1answer
63 views

Sequence of invertible operators converging in norm to an injective but not surjective operator?

Let $X$ be a complex Banach space. Can you find a sequence $A_n$ invertible and $A$ injective but not surjective in $B(X)$ such that $A_n\rightarrow A$ in norm? I know that such an $A$ must not be ...
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57 views

If $B$ is the restriction of an operator $A$ to an invariant closed subspace, is there an inclusion relation between $\rho(A)$ and $\rho(B)$?

Let $A$ be a linear operator on a $\mathbb R$-Banach space $E$ and $K$ be a closed subspace of $E$ such that $A$ is $K$-invariant, i.e. $$A(\mathcal D(A)\cap K)\subseteq K.\tag1$$ Under the given ...
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1answer
63 views

If $A$ is a contraction, then $(1-\lambda A)^{-1}=\sum_{n=0}^\infty\lambda^nA^n$ for all $|\lambda|<1$

Let $A$ be a self-adjoint contraction (operator norm at most $1$) on a $\mathbb R$-Hilbert space $H$. By contractivity the spectrum $\sigma(A)$ is contained in $[-1,1]$. Let $\lambda\in[0,1)$. How ...
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1answer
21 views

Bounded Linear “Change of Basis”

Let $E$ be a separable Banach space and $x,y \in E$. Does there necessarily exist a bijective bounded linear operator $A\in B(E)$ such that $$ Ax =y, $$ such that $A^{-1}$ is also a bounded linear ...
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7 views

Matrix reps of Operators acting on Nd states

I'm struggling trying to find a concrete (ie with numbers) example of a mat-representation of a differential operator that acts on 2(or more)-dimensional states. For 1d, it might look like this: $$ \...
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1answer
20 views

The restriction of a self-adjoint operator $A$ to ${\mathcal N(\lambda-A)}^\perp$ is symmetric, but is it even self-adjoint?

Let $H$ be a $\mathbb R$-Hilbert space, $A$ be a densely-defined self-adjoint linear oprator on $H$, $\lambda\in\mathbb R$ and $A_\lambda:=\left.A\right|_{\mathcal D(A)\:\cap\:{\mathcal N(\lambda-A)}^\...
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1answer
44 views

If $\lambda$ is an eigenvalue of a self-adjoint operator, is $\lambda$ in the resolvent set of $\left.A\right|_{{\mathcal N(\lambda-A)}^\perp}$?

Let $A$ be a symmetric linear operator on a $\mathbb R$-Hilbert space $H$ and $\lambda\in\mathbb R$. It's easy to see that $$A\left(\mathcal D(A)\cap{\mathcal N(\lambda-A)}^\perp\right)\subseteq{\...
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21 views

If $\lambda$ is an eigenvalue of a normal operator $A$, then $\left.A\right|_{{\mathcal N(\lambda-A)}^\perp}$ is well-defined and normal as well

Let $H$ be a $\mathbb R$-Hilbert space, $A$ be a normal$^1$ linear operator on $H$ and $\lambda\in\mathbb R$ be an eigenvalue of $A$. Are we able to show that ${\mathcal N(\lambda-A)}^\perp\...
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1answer
48 views

Trace class operators coming through the faithful representation of $L^\infty$.

Let $G$ be any locally compact Hausdorff group equipped with a $\sigma$-finite measure $\mu$. Consider the faithful representation $\pi:L^\infty(G)\to B(L^2(G))$ such that $\pi(f)(g)=fg$. Define $...
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42 views

Reference for Fredholm's alternative

i need a reference for my thesis. It concerns Fredholm's alternative in the following form: Let $\mathfrak{L}$ be a self-adjoint operator in a Hilbert space $H$ with a scalar product $\left< \cdot,...
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1answer
44 views

First countability strong operator topology.

Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{U}(\mathcal{H})$ be the group of unitary automorphisms, endowed with the strong operator topology. Now I want to show $\mathcal{U}(\mathcal{H})$ ...
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25 views

If every linear operator on a normed space bounded, is it possible to infer that the space is finite dimensional [duplicate]

It is well known that every linear operator on a finite dimensional normed space is bounded (continuous). The question that presents itself: is the reverse also true ? I mean: let $X$ be a normed ...
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1answer
25 views

Power Bounded Operator on Finite Dimensional Normed Spaces

Let $(\mathbb{X},||\cdot||_{\mathbb{X}})$ be a finite dimensional normed space (i.e, $(\mathbb{X},||\cdot||_{\mathbb{X}})$ is a banach space). Let $T: (\mathbb{X},||\cdot||_{\mathbb{X}}) \...
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9 views

Is the Laplacian corresponding to the self-adjoint operator induced by a Metropolis-Hastings Markov kernel on $L^2$ invertible?

Let $(E,\mathcal E,\mu)$ be a probability space and $\kappa$ denote the Markov kernel corresponding to the Metropolis-Hastings algorithm with target distribution $\mu$ (see, for example, $(1)$ in ...
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33 views

Closed form expression for the inverse of the Laplacian corresponding to the self-adjoint operator induced by a stationary Markov kernel on $L^2$

Let $(E,\mathcal E,\mu)$ be a probabiity space, $Q$ be a Markov kernel on $(E,\mathcal E)$, $\varrho:E\to(0,1]$ be $\mathcal E$-measurableand $$\kappa(x,B):=\varrho(x)Q(x,A)+(1-\varrho(x))1_A(x)\;\;\;\...
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1answer
30 views

If $A^{1/2}$ is the square-root defined in terms of the spectral decomposition, are we able to show that $\mathcal D(A^{1/2})\supseteq\mathcal D(A)$?

Let $H$ be a $\mathbb R$-Hilbert space, $A$ be a densely-defined nonnegative self-adjoint linear operator on $H$, $(\pi_\lambda)_{\lambda\in\mathbb R}$ denote the spectral family on $H$ corresponding ...
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23 views

Looking for an example for bounded operators

Let $T:X\to Y$ be a bounded operator, where $X$ and $Y$ are Banach spaces. Then could it happen that $\overline{TO_X(1)}\not\subset TO_X(2)$? Here $O_X(r):=\{x\in X:\|x\|<r\}$ for $r\in\mathbb{R}...
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1answer
28 views

Show that this parameterized family of operators is right-differentiable in its parameter

Let $H$ be a $\mathbb R$-Hilbert space, $\kappa^{(i)}$ be a linear self-adjoint contraction (i.e. a bounded operator with operator norm at most $1$) on $H$ and $\lambda\in(0,1)$. It's easy to see that ...
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18 views

Space of functions “dual” to $\mathcal{M}(X)$ under the operation of minimzation

I would like to prove/disprove that given $X$ finite and a space of probability measures $\mathcal{M}(X)$ on it (with weak* topology) there exists a subset $S(\mathcal{M}(X))$ of functions in $\mathbb{...
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22 views

Spectral radius and power boundedness

Let $(\mathbb{X},||\cdot||_{\mathbb{X}})$ be a (real or complex) Banach space. Let $(\mathbb{B}(\mathbb{X}),||\cdot||_{\mathbb{B}(\mathbb{X})})$ be the space of bounded linear operators of $\mathbb{...
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2answers
28 views

Increasing Sequence of projection $P_n \rightarrow 1_H$ then for any compact operator $K, \|[P_n,K]\|=\|P_nK-KP_n\| \rightarrow 0$

I want to prove that T is quasidiagonal iff $T=D+K$ where $D$ is a diagonal operator and $K$ is a compact operator. If $T=D+K$ then I want to show that $T$ is quasidiagonal. For this I want to show ...
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1answer
36 views

Definition of Inverse of Unbounded Operator

I am currently studying unbounded operators. However, in the text I am using, the definition for the inverse of an unbounded operator is given. I also went to several other books I own and the ...
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36 views

Polynomial maps of “close” operators (imitating continuity)

Let $T$ be a bounded operator on a Hilbert space. Let $L$ be another bounded linear operator satisfying $\|L\|<\varepsilon$. What can be said about the relationship between the spectra of $f(T),f(T+...
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19 views

What is an example of an antilinear operator

A linear operator $L$ is defined as $$L\Big(c_1f_1(x)+c_2f_2(x)\Big)=c_1Lf_1(x)+c_2Lf_2(x)$$ where $c_{1,2}$ are complex numbers. An example is $d/dx$. What could be an example of an antilinear ...
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14 views

Discrete maximum principle for for a discrete parabolic operator

While reading a paper on the topic 'Numerical solutions for generalized Black-Scholes equation', It is given that their numerical scheme can be executed explicitly by solving a linear system $\mathbf ...
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1answer
34 views

Operator norm of the self adjoint part

Let $A=\frac{1}{2}(T+T^\ast)$ be the self adjoint part of a bounded operator $T$ on a Hilbert space. Let $f$ be a polynomial. When do we have $\|f(A)\|\leq \|f(T)\|$ in the operator norm? The ...
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31 views

Conditions for a subalgebra to be weakly dense

Let $H$ be a complex, separable Hilbert space and $\mathcal{B}(H)$ denote the algebra of bounded linear operators on $H$. Let $A \subset \mathcal{B}(H)$ by a subalgebra. I'm looking for some ...
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27 views

Adjoint vs Self-adjoint operators represented by matrices

I want to see the difference between just adjoint and self-adjoint (hermitian) operator represented by matrices. If I have a matrix $$A= \begin{pmatrix} 1 & i \\ i & 1 \\ \...
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33 views

Dilation of a contraction in the connection with numerical range.

Let $T\in\mathcal{B(\mathcal{H})}$ be a contraction and $X\in M_n$ with $\Vert X\Vert\leq 1$ s.t. $W(X)\subseteq \overline{W(T)}$ where $W(T)=\{\langle Tx,x\rangle :\Vert x\Vert=1\}$ is the numerical ...
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1answer
44 views

Norm of Ratio of Operators

Suppose $a(L)$ and $b(L)$ are series of negative powers of the lag operator, that is $$a(L) = \sum_{j=1}^\infty a_jL^{-j},\quad\text{and}\quad b(L) = \sum_{j=1}^\infty b_jL^{-j}.$$ Also suppose that $...
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1answer
21 views

Intuition for absolute value of a bounded operator in the context of polar decomposition

Let $T$ be a bounded operator on Hilbert space. The functional calculus for bounded symmetric operators defines a positive symmetric $|T|=\sqrt{T^\ast T}$. Different operators can have the same ...
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2answers
67 views

When is $\mathbb{B}(\mathbb{X})$ compact?

Let $\mathbb{X}$ be a (real or complex) Banach Space and let $\mathbb{B}(\mathbb{X})$ be the space of all bounded linear operators of $\mathbb{X}$. Let $\tau_u$ be the Uniform Topology on $\mathbb{B}...
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1answer
69 views

Mathematical operations order when using an operator

I am not very familiar with operators (as I do not study mathematics) and I have just started a Quantum Mechanics course in a university. However, I am not sure what should be the precise order of ...
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1answer
26 views

two equivalent projections in matrix algebra

Let $A$ be a unital $C^*$-algebra and $a$ be a positive element in $A$ such that $\|a\|\leq 1$,show that $p =\left [ \begin{matrix} a & (a-a^2)^{\frac{1}{2}} \\ (a-a^2)^{\frac{1}{2}} &...
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0answers
19 views

What is the difference between compositional inverse and multiplicative inverse in Quantum mechanic?

Really am mixed , I have read a definition of bounded linear operator as it defined below A bounded linear operator $U: H \to H$ on a Hilbert space $H$ is called a unitary operator if it satisfies $...
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1answer
35 views

Schatten class operators form Banach algebra?

I am reading about Schatten $p$-class operators. Denote by $S_p(H)$ the space of all bounded linear operators with finite Schatten $p$-norm. I know that $S_p(H)$ is an ideal of $B(H)$ and is a Banach ...
2
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1answer
26 views

comparision of two positive invertible elements in a $C^*$-algebra

Suppose $A$ is a unital $C^*$- algebra,$a,b$ are any two positive invertible element in $A$,do there exist $s,t>0$ such that $sa\leq b\leq ta$?
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1answer
502 views

Does $\sigma(T) = \{1\}$ and $\|T\| = 1$ imply that $T$ is the identity?

Suppose that $T$ is a bounded linear operator on a complex Banach space X and that we know that $\sigma(T) = \{1\}$ and $\|T\| = 1$ (i.e. the spectrum of the contraction $T$ consists only of a single ...
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30 views

Is this (Hibert-Schmidt) integral kernel bounded?

It is well known in the theory of bounded operators on $L_2(\mathbb{R}, \mu)$ that the operator $$ Tf(x) = \int_\mathbb{R} k(x,y) f(y) \: dy $$ is compact whenever $k(x,y) \in L_2(\mathbb{R}\times \...
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0answers
36 views

Infinitesimal generator of the average absolute deviation of Poisson process

I am trying to find the infinitesimal generator of the average absolute deviation of the Poisson process. i.e. the generator of $X_n(t)=\dfrac{Y(n^2t)-\lambda n^2t}{n}$. Observe that even though it'...