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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Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$?

I am a beginner of funcional analysis. I have a simple question when I study this subject. Let $L(X)$ denote the Banach algebra of all bounded linear operators on Banach space X, $T\in X$ is ...
2
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51 views

I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
2
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1answer
36 views

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 $X$, ...
2
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1answer
66 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]) \...
2
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0answers
17 views

Toeplitz operators on $\ell_p$ modulo compact operators

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 ...
6
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1answer
75 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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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 $V:=\iota(U)$ ...
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49 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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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 $...
1
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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' ...
3
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0answers
37 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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0answers
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 ...
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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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 ...
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25 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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0answers
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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0answers
19 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 }...
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0answers
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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 ...
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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2answers
103 views

Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$ Some definitions and results of the ...
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33 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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1answer
16 views

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 ...
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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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39 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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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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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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0answers
34 views

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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1answer
21 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 ...
5
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1answer
101 views

Connection of Fourier's work with Fredholm's

Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with ...
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0answers
38 views

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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2answers
48 views

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 ...
5
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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 ...
2
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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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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 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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Is probability function for mutually exclusive events a linear operator?

If the definitive classification criteria for a linear operator are given by: $L(f+g) = L(f) + L(g)$ [for any/every pair of functions, $f$ and $g$] $L(tf) = tL(f)$ [for ...
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1answer
56 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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1answer
47 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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2answers
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Why is $\overline{B(l^2)\odot B(l^2)}^{\| \enspace \|_{op}}\neq B(l^2\otimes l^2)?$

Let $B(l^2)$ be the $C^*$algebra of bounded linear operators on the sequence space $l^2$ and denote with $B(l^2)\odot B(l^2)$ the tensor product of $B(l^2)$ with itself, considered as a $*$algebra (...
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1answer
79 views

Prob. 9, Sec. 3.10 in Kreyszig's functional analysis book: The image of ann isometric non-unitary operator on a Hilbert space

Let $H$ be a Hilbert space, let $T \colon H \to H$ be a linear operator such that $T$ is isometric but not unitary. Then how to show that the image $T[H]$ is a proper closed subspace of $H$? My ...
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2answers
50 views

Prob. 8, Sec. 3.10 in Kreyszig's functional analysis book: An isometric linear operator has its adjoint as its left inverse

Let $H$ be a Hilbert space, and let $T \colon H \to H$ satisfy $$\langle Tx, Tx \rangle = \langle x, x \rangle \ \mbox{ for all } \ x \in H.$$ Then $T$ is bounded and norm $\Vert T \Vert = 1$ (unless ...
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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 ...
2
votes
1answer
46 views

projections in von Neuman algebra

Consider a semifinite von Neumann algebra $\mathcal{M}$ with a semifinite faithful normal trace $\tau$. If $Q, P$ are projections in $\mathcal{M}$ with $\tau(Q)< \tau(P)$, then does $\tau(P\wedge Q^...
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0answers
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]\...
2
votes
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
25 views

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 ...
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0answers
35 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(\...