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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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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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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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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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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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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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 ...
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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 ...
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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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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 ...
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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
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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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}(\...
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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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44 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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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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22 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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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|\...
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1answer
33 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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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(\...
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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
48 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
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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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 ...
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Orbits under small perturbations

Suppose $T$ is a bounded linear operator on $l_2$ and $x\in l_2$ is a vector such that the orbit $(T^{n}x)$ is linearly independent. Can one find an $\epsilon>0$ such that for all $||y-x||<\...
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1answer
29 views

Are $X'\otimes Y$ and $\mathfrak L(X,Y)$ isomorphic?

Let $\mathbb F\in\left\{\mathbb C,\mathbb R\right\}$ $X$ and $Y$ be normed $\mathbb F$-vector spaces $X'$ be the topological dual space of $X$ $\mathfrak L(X,Y)$ be the set of bounded, linear ...
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1answer
26 views

range of weighted shift operator

Consider the weighted shift operator on $\ell^2$ space defined by $T(x_0, x_1, x_2, ...) = (0, x_0, 2x_1, 3x_2, 4x_3, ...)$ with domain $$\mathcal{D}(T) = \{(x_n) \in \ell^2 : \sum_{n=0}^{\infty}|(n+...
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proof of an equality norm

Let the mapping $T:\ell^{2}\rightarrow \ell^{2}$ is defined as follow. $$T(x_1,x_2,\ldots,x_n,\ldots)=(x_1,\dfrac{1}{2}x_2,\ldots,\dfrac{1}{n}x_n,\ldots)$$ In this case, i've easily earned: $$\sigma(T)...
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What does 'mode' mean in this context?

The spectrum of an operator $\mathcal{L}$ is the disjoint union of two sets: the point spectrum that consists of all isolated eigenvalues with finite multiplicity and its complement, which we call the ...
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1answer
78 views

Determine whether the differential operator is compact in the following cases

Given the differential operator $\displaystyle Tx(t)=\frac{dx}{dt}$, I need to determine (and be able to justify) whether it is compact in the following three cases: $T: C^{1}[0,1]\mapsto C[0,1]$ $T:...
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unitary operator between two Hilbert subspaces

$H$ is a Hilbert space. $P, Q$ are projections. For every $x\in P(H)$, we have decomposition $x = Qx +Q^\perp x$. Then, can we find a unitary operator from the space generated by all $Qx$, $x\in P(H)$...
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48 views

Diagonal operators on infinite dimensional Hilbert spaces

the following is a short question regarding a theorem from a quantum mechanics book I am working through but the question is a mathematical one. There is a theorem which states: Theorem: The ...
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Complex Air operator

Help me to do this exrcice Consider the differential operator $A=-\partial^{2}_{x}-ix$ on $\mathbb{R}$ with $D(A)=\{f\in L^{2}(\mathbb{R},dx), Au\in L^{2}(\mathbb{R},dx)\}$ Check that $A$ is colsed ...
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Is the generated semigroup by an elliptic operator be the transition semigroup?

I am considering the time homogeneous Ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dW_t,$$ where $b,\sigma$ currently are only assumed to be global Lipschitz for the existence of solution $X_t$. The ...
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2answers
36 views

why the quotient space is finite $X/\ker T$

Let $T:X\rightarrow Y$ be a linear operator from Banach space to Banach space, if $Y$ is finite dimensional, show $X/\ker T$ is finite dimensional, moreover has same dimension with $Y$. Any help is ...
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1answer
57 views

This linear operator has no eigenvalues

Let $T : L^2(\mathbb R) \to L^2(\mathbb R)$ be a linear operator defined by $$(Tf)(x)=f(x+1).$$ Show that $T$ has no eigenvalues, i.e., there exists no $f \not= 0$ in $L^2(\mathbb R)$ such that $(Tf)(...
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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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Existence of some extension

Let $X_{0}$ be a linear closed proper subspace of real normed space $X$. Show that for every linear and continuous functional $\phi_{0}: X_0 \to \mathbb{R} $ with norm 1 there exist a linear and ...
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1answer
52 views

Using calculus results for functions of operators

I am interested in the conditions required for functions of operators to be manipulated as if it were a real valued function with a real domain. In an applied maths text I am using the following is ...
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1answer
73 views

How can we prove that the space of trace class operators on a Hilbert space $H$ is the closure of $H\otimes H$ with respect to the trace norm?

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space over $\mathbb R$ $\mathfrak L^1(H)$ be the space of trace class operators on $H$ and $$\operatorname{tr}L:=\sum_{n\in\mathbb ...
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83 views

A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm diag}(0,0,\cdots,0,1,1,1,1,\cdots)$$...
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What are the Hermitian idempotent matrices with respect to $l_{1}$ norm

Let $A=(M^{n}(\mathbb{C}), \|\cdot\|_{1})$. The numerical range of $a\in A$ is defined as $V(a)=\{f(a):f\in A',\|f\|=1=f(1)\}$. $a\in A$ is said to be Hermitian if $V(a)\subseteq \mathbb{R}$. $a\in A$ ...
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How to generalize this proof of the closed graph theorem

I found this tricky new proof of the closed graph theorem for a Hilbert space $H$. http://arxiv.org/pdf/1601.02600.pdf It says in the abstract, that it's possible to extend the proof to Banach space. ...
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Metrics from Operator Norms

Let $X$ be a Hilbert space and $(\cdot,\cdot)_X$ be the inner product on $X$. It is well known that $|x|_X = \sqrt{(x,x)_X}$ is a norm on $X$ and $|x-y|_X$ is a metric on $X$. The norm on $X$ induces ...
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49 views

Norm of operator $A$ st. $A^2 = I$?

I'm wondering what can be said about the norm $||A||$ of an operator which squares to identity. All I can think of is that $$1=||AA|| \leq ||A||^2$$ so that $||A|| \geq 1$. But can anything else be ...
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1answer
38 views

spectral projection of an isolated point in the spectrum of a closed linear operator

Suppose that $T$ is a closed densely defined operator on a Hilbert space $H$ with $\rho(T) \neq \emptyset$. If $\lambda \in \sigma(T)$ is an isolated point then we know that $H = \mathcal{N}(E_o) \...
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1answer
40 views

Weak convergence and strong convergence on $B(H)$

Let $\mathcal{A} \subset B(H)$ be a weak closed convex bounded set of self-adjoint operators. If $A_n \rightarrow_{wo} A\in \mathcal{A}$, do we have $A_n \rightarrow A$ strongly?($A_n$ is a sequence ...
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1answer
17 views

In search of a necessary condition for completeness of some metric space with application to pde

$A$ is an operator. Consider a metric space $K$ (a function $f$ is in $K$ if and only if $Af$ is in $L^2$) where the metric between two functions $f$ and $g$ is defined as $\mu (f ,g) = \int_{R^3} (...
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1answer
29 views

Closedness of first order differential operator on $L^2(\Omega)$

I am considering the when the following first order differential operator is a closed operator $$Au=b(x)\dfrac{\partial u}{\partial x_i},$$ on $L^2(\Omega)$ with the domain $D(A)=H^1(\Omega)$. Here I ...