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 an operator restricted to an invariant subspace

Let $X$ be an infinite-dimensional real Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. Suppose $W$ is a finite-codimensional $T$-invariant closed subspace of $X$, ...
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1answer
36 views

Let $H$ be a Hilbert space, $V≤H$ be closed, $Q:H→V$ be the orthogonal projection, $(e_n)_{n∈ℕ}$ be an ONB of $H$. Is $(Qe_n)_{n∈ℕ}$ an ONB of $V$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $H$ be $\mathbb K$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $\iota:U\to H$ be an embedding and $V:=\iota(U)$ ...
2
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1answer
62 views

Find the spectrum of an operator

I am trying to learn some basic stuff about spectral theory, and I am a little bit lost. Please, could you help me and tell me how to find $\sigma(T)$ and $\sigma_p(T)$ of the operator $T:C([0,1]) \...
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1answer
31 views

Linear Operator on Hilbert Space $l(\mathbb Z)$

Let $A$ be the linear operator on $l(\mathbb Z)$ defined for $u=\{u_k\}_{k \in \mathbb Z}$ as $(Au)_k = \sum_{h=-\infty}^{+\infty}a_{k,h}u_h$ where $a_{k,h}=\frac{1}{(k-h)2}$ for $h \not= k$, and $...
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46 views

Linear Operators on $L_2(\mathbb R)$ definfed as Integrals

Let's consider the linear operators on $L_2(\mathbb R)$ $$ T_{\alpha}f(x) = \int_{-\infty}^{+\infty} \frac{e^{-|x-y|^2}}{(1+x^2)^{\alpha}}f(y)dy $$ with ${\alpha} \in [0,1]$. Find ${\alpha}$ such ...
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36 views

Square-root of $\iota\iota^\ast$, where $\iota$ is an isometric embedding between Hilbert spaces

Let $U$ and $H$ be Hilbert spaces and $\iota$ be an embedding of $U$ into $H$. Then, $$\pi x:=u\;\;\;\text{for }x\in H\text{ with }x=\iota u+y\text{ for some }u\in U\text{ and }y\in\left(\iota U\right)...
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1answer
35 views

Dimension of operators

Question: Let $T: \ell^2 \to \ell^2$ be self-adjoint and compact. For $\lambda \in \mathbb{R}$, let $$ S_\lambda = \overline{\text{Span} \{ v \in \ell^2 \mid Tv = \gamma v \text{ for some } \gamma \le ...
2
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1answer
24 views

The point spectrum and residual spectrum of an operator on $l_2$ related to backward shift

I have a problem with the spectrum of this operator: $(Tx)_1 = x_2$ $(Tx)_2 = x_1$ $(Tx)_n = \frac{1}{n}x_{n+1}$ with $n\ge3$ Find the $||T||$, the point spectrum $\sigma_P(T)$ and $\sigma_P(T^{\...
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24 views

Nullspace of strange operator

I have the following equation: $0=\frac{\partial}{\partial y}(e^{-\beta U(x,y)}\frac{\partial}{\partial y}(P(x,y,t)e^{\beta U(x,y)}))$ and would like to study its solvability (Fredholm) conditions (...
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30 views

Given a special Hilbert space $U_0$, is there a proper superspace $V$ such that the inclusion $\iota:U_0\to V$ is Hilbert-Schmidt?

Let $U$ be a Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\rangle_U\;\;\;\text{for }u,v\in ...
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1answer
33 views

Can we find a concrete representation of $\iota\iota^\ast y$, if $\iota$ is a Hilbert-Schmidt embedding between Hilbert spaces?

Let $U$ and $H$ be real Hilbert spaces $\iota:U\to H$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ Can we find a concrete representation of $Qy$ for some $y\in H$? By Riesz' ...
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27 views

Complemented Spaces: Continuity vs. Closedness

Given a topological vector space $X$. (Not necessarily Hausdorff!) Consider subspaces: $$U_\pm\leq X:\quad X=U_++U_-\quad U_+\cap U_-=(0)$$ Equivalently an isomorphism: $$\Phi:U_+\oplus U_-\...
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1answer
28 views

Closable Operators: Nonexample

Given the Banach space $X:=\mathcal{C}([0,1]\cup[2,3])$. I remember I've seen a beautiful example of a non-closable operator whose graph is dense. It involved exploiting Stone-Weierstraß for a ...
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1answer
22 views

Hahn-Banach: Operators

Given two Banach spaces $X$ and $Y$. (More generally locally convex spaces) Regard a closed subspace $U\subseteq X$. Does every bounded operator extend: $$T\in\mathcal{B}(U,Y)\implies T_E\in\mathcal{...
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18 views

If $U_0,V$ are Hilbert spaces, $(e_n)$ is an ONB of $U_0$ and $ι:U_0→V$ is an embedding, can we complete $(ιe_n)$ to an ONB of $V$?

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\rangle_U\;\;\;\text{for }...
4
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1answer
33 views

Is the Null Space of an linear operator the same with the Null Space of its associated hermitian?

Let A be a bounded linear operator on $H$ where $H$ is a (not necessary I think, but in my case separable) Hilbert space. Then, the question: is its null space the same as the null space of the ...
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Did I make mistakes? Bilinear form, generator, strange relation

I have a question about functional analysis and operator theory. Definition Let $(H,(\cdot,\cdot)_{H})$ be a real Hilbert space and $D$ be a dense subspace of $H$. Let $(\mathcal{E},D)$ be ...
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32 views

Accreative operator in Banach space

Let $X$ be a Banach space and $H$ be a Hilbert space such that $X$ is embedded in $H$. By a duality map, $\phi_{x}$, defined on $X$, we mean any linear functional $$\phi_{x} \in \left\{f \in X'\ \...
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invertible operators in Banach spaces [closed]

Let $X$ be a Banach space Take $V$ a positive operator such that $V\geq I_{X}$. Can we prove that $V$ is an invertible operator?
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2answers
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Positive-definite 2x2 matrix invariant under conjugation with W -> W unitary?

I have the following problem for complex matrices: $G$ is a positive-definite, hermitian 2x2-matrix and $W$ an invertible 2x2 matrix. Moreover, $WGW^\dagger = G$ holds. Does this imply that $W$ is ...
2
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1answer
29 views

Show that $T^{-1}:Y \to X$ exists and is bounded.

Let $T$ be a bounded linear operator from a normed space $X$ onto a normed space $Y$. If there is a positive $b$ such that $$||Tx||\ge b||x||,$$ for all $x \in X$, show that $T^{-1}:Y \to X$ exists ...
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36 views

Norm of an operator in a Hilbert space

Let $T\neq 0, \neq I$ be a linear operator of a Hilbert space such that $T \circ T = T $. Show that $\|T\|=\|I-T\|$. Anyone has an idea ? I just proved that $\|I-T\| \leq 1 + \|T\|$ but it is not ...
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1answer
73 views

Trace term in the Itō formula

I'm reading Stochastic Differential Equations in Infinite Dimensions and don't understand what the authors do in Chapter 2.3.1. Let me introduce the necessary objects: Let $K$ and $H$ be real ...
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0answers
21 views

Spectrum of $\sigma^{-1/2}\rho\sigma^{-1/2}$

Let $\rho$ and $\sigma$ be trace class operators. Thus, they have eigenvalues and eigenvectors, and can be diagonalized just as finite dimensional matrices. Now we consider $\sigma^{-1/2}\rho\sigma^{-...
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23 views

Do you know an operation in which the order of elements does not matter but repetitions is allowed?

In order to present a concept in an article, I need an operation in which the order of elements does not matter but repetitions is allowed. For example, if a have some vectors [1 2 3], [2 3 1], [1 2 ...
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I don't understand how the adjoint operator is used in a book that I'm reading

I'm reading Stochastic Differential Equations in Infinite Dimensions and don't understand what the authors do in Chapter 2.3.1. Let me introduce the necessary objects: Let $K$ and $H$ be real ...
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the imageof the closed unitball of a Hilbert space under a compact linear map between Hilbert spaces, is compact

Could you plz give me a hand in this problem? Thanks in advanced let T: H_1→H_2 be a compact linear map between Hilbert spaces H_1 and H_2. then the image of the closed unit ball of H_1 under T is ...
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If $Q$ is a trace class operator on $U$, then each bounded, linear operator from $U$ to $H$ is a Hilbert-Schmidt operator from $Q^{1/2}U$ to $H$

Let $U$ and $H$ be Hilbert spaces $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace $U_0:=Q^{1/2}U$ $L$ be a bounded, linear operator from $U_0$ to $H$ I ...
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1answer
31 views

How to show that this integral operator is bounded?

Consider the integral operator $T : C([0,1])\to C([0,1])$ given by $$Tf(t)=\int_0^1 K(t,\tau)f(\tau)d\tau.$$ I'm solving one exercise which is to show this operator is bounded. The exercise is from ...
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1answer
35 views

Operator with norm

I got the following problem to solve: Let $H$ Hilbert space and $T: H \to H$ a bounded positive operator, i.e. \begin{align*} \langle x, T x \rangle \geq 0 & & \text{for all } x \in H. \end{...
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2answers
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finite spectrum eigenvalue

Let $T:X \to X$ be a linear bounded operator where X is Banach space ,and $\sigma (T)$ is a finite set.Then does the spectrum consist of eigenvalues only? Any hint or counterexample is ok. thanks in ...
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60 views

Can we use sequences to test continuity of a weak$^*$-continuous operator?

Let $X,Y$ be Banach spaces. Now assume we have a map $T:X'\rightarrow Y'$ where $X'$ and $Y'$ are equipped with the weak$^*$ topology and not the norm topology. Can I infer from this that an operator ...
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1answer
46 views

Is $Ax = (\alpha_1 x_1, \alpha_2 x_2, \alpha_3 x_3, \dots, \alpha_k x_k, 0, 0, \dots)$ a compact operator?

Is the operator $A$ defined by $$Ax = (\alpha_1 x_1, \alpha_2 x_2, \alpha_3 x_3, \dots, \alpha_k x_k, 0, 0, \dots)$$ a compact operator? It only has finitely non-zero dimensions, so does this mean ...
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1answer
20 views

Convergence of unitary products on a Hilbert space

First: I'm sorry for the basic question--I can move it to Math SE if necessary... Let $X$ be a Hilbert space and suppose $\{U_k\}_k$ is a sequence of unitary operators on $X$. Let $\|\cdot\|$ be the ...
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1answer
43 views

Eigenfunctions of an integral operator

Let $Tf(x):=\int_0^x f(t)dt$ be an integral Operator ($T:L_2[0,1]\rightarrow L_2[0,1]$). I am trying to find the eigenvalues and eigenfunctions of $S:=T^*T:L_2[0,1]\rightarrow L_2[0,1]$. So far I know ...
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15 views

Ask for reference convergence of implicit euler method for initial value problem with dissipative source term

I am considering the convergence of implicit euler method for solving the following initial value problem: \begin{cases} u'(t)=f(t,u(t)),t\in[0,T]\\ u(0)=u_0\in \mathbb{R}, \end{cases} where $u:[0,T]\...
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12 views

Subnormal Weighted shift and First order derivative

Let $\mathbb B^m$ denote the Eucledian ball in $\mathbb C^m.$ Does there exist a reinhardt measure $\mu$ supported on $\partial \mathbb B^m,$ the boundary of ball, so that the Hilber space $H^2(\mu)$, ...
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1answer
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How to show $\sigma(T_q) = \overline{\{q(t) : t \in [0,1]\}}$ where $T_q$ is the multiplication operator?

Let $B$ be the Banach space of bounded complex functions on $[0,1]$ with sup-norm. For $q \in B$, define the (multiplication) operator $T_q : B\rightarrow B$ by $(M_q f)(t) = q(t)f(t)$. How do you ...
3
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1answer
150 views

Show that if $[Q,P]=it\Bbb{I}$ then the operators are unbounded

In the Hilbert space $\mathcal{H} = L^2(\mathbb{R},dx)$, let 2 symmetrical operators $P$ and $Q$ be given, with the following properties: $D(P) = D(Q) = \mathcal{S}(\mathbb{R})$ $P\mathcal{S}(\...
2
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1answer
46 views

What does a homomorphism $\phi: M_k \to M_n$ look like?

Let $\phi : M_k(\Bbb{C}) \to M_n(\Bbb{C})$ be a homomorphism of $C^*$-algebras. We know that $\phi$ decomposes as a direct sum of irreducible representations, each of them equivalent to the identity ...
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1answer
42 views

Spectrum of linear operator, essential spectral radius

Consider the operator $L:L^1(S^1)\to L^1(S^1)$ given by $$ (Tf)(x)=\dfrac{1}{2}\left( f\left( \dfrac{x}{2}\mod 1\right)+f\left( \dfrac{x+1}{2} \mod 1 \right) \right) $$ where we identified $S^1$ with ...
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1answer
67 views

Representation of the Fréchet derivative of $〈f,e_n〉$, where $f:H→H$, $H$ is a Hilbert space and $(e_n)_{n∈ℕ}$ is an orthonormal basis of $H$

Let $H$ be a $\mathbb R$-Hilbert space $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $f:H\to H$ be Fréchet differentiable and $$f_n:=\langle f,e_n\rangle\;\;\;\text{for }n\in\mathbb N$$ $\...
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1answer
20 views

About properties of invertible operators

Let $X$ be a Banach space and $S\in\mathcal{L}(X)$ be a bounded invertible operator. Take $X_{1}$ and $X_{2}$ two subspaces of $X$ such that $X_{1}\subseteq X_{2}$ and consider $S: X_{1}\rightarrow ...
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1answer
25 views

Resolvent $R(\lambda,A)x \to 0$ as $|\lambda| \to \infty$

If I have a closed operator $A:D(A) \to X$, not necessarily bounded on a Banach space $X$, and the resolvent is unbounded, can I show for a fixed $x \in X$ that $$R(\lambda,A)x \to 0$$ as $|\lambda|\...
0
votes
1answer
32 views

Properties of Injective Operator on Hilbert Space

I am new to functional analysis and have the following issue: Given an infinite dimensional Hilbert space $H$ and an operator $f: H \times \Omega \to H$, where $\Omega$ is some finite dimensional ...
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0answers
31 views

Schmidt decomposition problem

I'm having a problem in implementing the following problem: I have a quantum state so defined: $\left| \Psi\right>=\int \mathcal{A}(\omega_1,\omega_2)\hat{a}^\dagger_H(\omega_1)\hat{a}^\dagger_V(\...
0
votes
1answer
38 views

Why are all linear maps on $\mathbb{R}^d$ bounded?

Suppose $W$ is a normed space and $A : \mathbb{R}^d \to W$ is a linear operator. Why is $A$ automatically bounded?
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1answer
47 views

An expression for the Hilbert-Schmidt inner product

Suppose that $k:[0,1]\times[0,1]\to\mathbb C$ is a Hilbert-Schmidt kernel, i.e. $$ \int_0^1\int_0^1|k(x,y)|^2\mathrm dx\mathrm dy<\infty. $$ The associated Hilbert-Schmidt integral operator $K:L^2([...
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1answer
55 views

Can we embed $X'\otimes Y$ into the space of bounded, linear operators $X\to Y$?

Let $\mathbb F\in\left\{\mathbb C,\mathbb R\right\}$ $X$ and $Y$ be normed $\mathbb F$-vector spaces $X'$ denote the topological dual space of $X$ $\mathfrak L(X,Y)$ denote the space of bounded, ...
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0answers
17 views

How to prove that $ e_{\lambda}$ can be written in the following form?

Let $e_{\lambda}$ be the spectral density associated to the spectral function $E_{\lambda}$ for a self-adjoint operator $A$ on a complex Hilbert space $(H,\left<., .\right>)$. Haw to prove that ...