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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Spectral Measures: Normality

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
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Two normal operator that commutes

Suppose $N\in B(H)$ is normal, and $T\in B(H)$ is invertible. Prove that if $TNT^{-1}$ is normal then $N$ commutes with $T^*T$. I can not any idea to prove it, just I know ...
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Prove that a functional is convex

Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. We define the functional $\Phi$ as: \begin{equation} \Phi(x)=\frac{1}{2}(Tx,x) \end{equation} My exercise says that $\Phi$ is ...
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32 views

Frechet derivative of an operator

Let an operator $T:C[a,b]\to C[a,b]$ be defined as: \begin{equation} (Tu)(x)=\int_{a}^{b}K(x,t)f(t,u(t))dt \end{equation} where $K:[a,b]\times[a,b]\to \mathbb{R}$, $f:[a,b]\times \mathbb{R}\to ...
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24 views

Spectral Measures: Boundedness

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
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Normal Operators: Transform (III)

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ By the previous threads: $$N=Z\sqrt{1-Z^*Z}^{-1}$$ Especially one had: ...
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Spectral Measures: Invertibility

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
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Mourre Adjoint: Bounded Maps (III)

I will provide an answer later... Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And an operator: ...
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What is the relation between the matrix of a bounded linear operator and that of its adjoint operator?

Let $H_1$ and $H_2$ be finite-dimensional (real or complex) Hilbert spaces, let $T \colon H_1 \to H_2$ be a linear operator, [Then $T$ can be shown to be bounded] and let $T^* \colon H_2 \to H_1$ ...
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How to prove Cholesky decomposition for positive-semidefinite matrices?

According to Cholesky decomposition $A$ is a Hermitian positive-definite matrix if and only if $A=T^*T$ for some upper triangular matrix $T$. When $A$ is positive-semidefinite we have such ...
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Question about trace class operators

Let $\cal{H}$ be a Hilbert space, $T$ a bounded linear operator on $\cal{H}$, $S$ a trace class operator, then can one verify that $$|Tr(TS)|\leq\|T\|\cdot|Tr(S)|?$$
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Dimension of a subspace of a Hilbert space

Suppose $\{A_n\}$ is a sequence of operators on Hilbert space $H$ such that $\dim (\overline{\operatorname{Im} A_n})\leq \alpha$ where $\alpha\geq N_0$. If $A_n\to A$ uniformly, then $AH=\lim A_n ...
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Construct a set of anticommutative operators

Suppose that I have a set of infinitely many operators $\mathbb{A} = \{ A_1, \dots , A_n \}$ with $n \mapsto \infty$. All operators satisfy $A_i A_j + A_j A_i = 0$ for all $i,j \in \{ 1, \dots, n ...
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C*−algebras and operator theory, murphy [duplicate]

Do you know how to solve this exercise? (Murphy, $C*$−algebras and operator theory, $2^{nd}$ chapter, 1st exercise) Let $A$ be a Banach algebra such that for all a\in A the implication $Aa=0$ or ...
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Find the inverse of an operator, and determine is it bounded.

I've been doing some similar problems, but I got stuck on this one... and I have a feeling I'm running in circles trying to solve it. Any help appreciated! Problem: We have an operator: $$ A : ...
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A problem which reverses the definition of a bounded operator

I've encontered a problem that appears simple, almost like it's a definition of a bounded operator, but with a reversed inequality sign... and I can't seem to find my way to a solution. Any ...
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1answer
41 views

Mourre Adjoint: Bounded Maps (II)

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the domain: ...
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Fractional powers of positive self-adjoint operators

Consider two positive unbounded operators $A$ and $B$ densely defined on a Hilbert space $H$ self-adjoint on a domain $\mathcal{D}(A) = \mathcal{D}(B) = H_1$. By the spectral theorem, we can define ...
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49 views

Mourre Adjoint: Bounded Maps (I)

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the domain: ...
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Pureness of Vector States

How does one show that irreducibility is equivalent to a vector state being pure? In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let ...
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${x_n} \to x$ weakly, why does $T{x_n} \to Tx$ weakly? [closed]

If $T \in B(X,Y)$ and ${x_n} \to x$ weakly, why does $T{x_n} \to Tx$ weakly?
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ODE's & PDE's: Homogenous piecing vs Eigenexpansion vs Green functions

I don't know if i'm within rules of the forum to ask this question. If i'm not please comment before downvoting. If you know of a source that answers these questions, please suggest. It would be ...
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Quick question about the Frobenius method

When solving the the eigenvalue equation $\mathcal{L}\phi = E\phi$, where $\mathcal{L} = \left \{-\frac{d^2}{dx^2} + x^2 \right \}$ is a Sturm-Liouville operator, using the Frobenius Method $\phi = ...
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Generating fractional taylor series

I was considering the notion of taylor series which posit that the sum $$ \sum_{i=0}^{\infty} \frac{1}{i!} a_ix^i $$ Where: $$ a_i = \frac{d^if}{dx^i}_{x= a} $$ Converge to the function f in a ...
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Riesz representation theorem for $\langle\mathcal A u,v\rangle$.

Let $V$ be a Hilbert space, and let $V^*$ denote its dual space, consisting of all continuous linear functionals from $V$ into the field $\mathbb R$ or $\mathbb C$. If $x$ is an element of $V$, then ...
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Spectral Measures: Constructions

Any constructions are welcome!!! Given a Hilbert space $\mathcal{H}$. Denote projections by: $$\mathcal{P}(\mathcal{H}):=\{P\in\mathcal{B}(\mathcal{H}):P^2=P=P^*\}$$ Consider spectral measures: ...
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Norm of the operator $T:\ell^2 \to \ell^2$ defined as $(Tx)_1=0, (Tx)_n=-x_n+\alpha x_{n+1}$

Consider the operator $T: \ell^2 \to \ell^2$ defined as $$\begin{cases} (Tx)_1 = 0, \\ (Tx)_n = -x_n + \alpha x_{n+1}, \quad n\ge 2 \end{cases} $$ where $\alpha \in \mathbb{C}$. I want to find ...
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Triangular projection on kernels of trace class operators

Let $k$ be a kernel of a trace class integral operator on $L^2(0, 1)$, and assume that $k(x, y)=-k(y, x)$. Define $l(x, y)=k(x, y)$ whenever $x>y$, and $l(x, y)=0$ if $x<y$. Does the integral ...
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x-momentum operator $p_x$ expressed as multiple of Translation operator

On this page https://en.wikipedia.org/wiki/Rotation_operator_%28quantum_mechanics%29 under "The translation operator," they use Taylor expansion. As part of that proof they state $p_x = ih * ...
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Norm of an integral operator in $L^1\bigl( (0,1)\times (0,1) \bigr)$

Let $I := (0,1)\times (0,1)$ and $q\colon I \to \mathbb{R}$ be a non-negative measurable function such that for all $x \in (0,1)$ we have $$ \int_0^1 q(x,y) dy = 1. $$ We denote by $L^1(I,ydxdy)$ the ...
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Every non-compact Hermitian operator P has an infinite dimensional invariant subspace on which P is bounded from below

I want an explanation of the following statement. If $P$ is a Hermitian operator on Hilbert space and not compact, there exists an infinite-dimensional subspace $M$, invariant under $P$, on which $P$ ...
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Can you have an operator on a vector space such that it is injective but its kernel is not the zero element?

Take any vector space $V$ and an operator $T : V \mapsto V$ Can there exist a $T$ such that it is injective but $\ker T \neq \{0\}$ and equal to some other element instead?
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uniformly convergent subsequence of bounded linear operators on a Hilbert space?

I am working a problem in which we start with a Hilbert space $\mathcal{H}$ and a sequence $\left\{a_n\right\} \subset \mathcal{H}$ with $||a_n|| = 1$. We also assume that $$\lim_{n \to \infty} ...
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1answer
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The real version of the Cuntz algebra

Assume that $H$ is a real separable Hilbert space. Are there two operators $T,S \in B(H)$ which satisfy $$TT^{*}+SS^{*}=1,\;\;T^{*}T=S^{*}S=1$$ where * is the adjoint operator?
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Unboundedness of Laplacian

I am currently considering the following operator ("modified Laplacian"): $T \colon \left( W^{2,2}(\mathbb{R}), \| \cdot \|_{L^2} \right) \longrightarrow \left( L^2(\mathbb{R}), \| \cdot \|_{L^2} ...
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T is invertible if and only if $||(T_{n})^{-1}|| \leq M$.

Let $X$ be a Banach space. Let $(T_{n})$ be a sequence of invertible operators in $B(X)$ and let $T \in B(X)$ be the uniform limit of $(T_{n})$, i.e., $||T_{n}-T||\rightarrow0$ as $n \rightarrow ...
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Infer $Tf=\sum_{n=1}^\infty (f,f_n) f_n$ from frame condition.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that, for $A<B$ and for all $f\in H$ ...
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Eigenvalue-eigenvector equation for an operator

Proof: Given an eigenvalue-eigenvector equation, suppose that the state vector depends on an external parameter, e.g. time, and that over it acts an operator that is the fourth derivative w.r.t. ...
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1answer
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The (un)boundedness of an involutive operator

This is a question I've been thinking about for a while, for which I do not have a satisfactory answer. Suppose that $T$ is a densely-defined operator on a Hilbert space $\mathcal{H}$ such that ...
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Calculating Norms for the Transpose

How does one show that the transpose mapping $T: \mathcal{M}_{n} \to \mathcal{M}_{n}$, given by $T(a)=a^{t}$, has norm $||T||=1$ but completely bounded norm $||T||_{CB}=n$? Notation. ...
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Purely nondeterministic weakly stationary processes

I found a necessary and sufficient condition for a stochastic process being purely nondeterministic in Ihara (1993). As follows: A weakly stationary process $X$ is purely non-deterministic if and ...
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C*-algebra generated by a non-invertible element

Let $x$ be a non-invertible element, and put $A:=C^*(x)$. Let $I$ be a closed ideal of $A$, and $\pi: A\to A/I$ be a natural quotient map. Is it possible that there is an invertible element $y\in A$ ...
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Wave Operators: Reducibility

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_0:\mathcal{D}(H_0)\to\mathcal{H}_0:\quad H_0=H_0^*$$ $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ and a bounded ...
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A particular decomposition of a CPTP map

Let $\mathfrak{D}$ denote the set of $n\times n$, trace-one, positive semi-definite matrices (known as density matrices in quantum information theory). Consider a Completely Positive Trace Preserving ...
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Change of variables for integral operator

One can write the operator $L=(\sqrt{1-i\partial_x^2}-1)$, as an integral, that is $$(\sqrt{1-i\partial_x^2}-1)B(x,t)=\frac{i}{4\pi^2} \int_{-\infty}^{\infty}(\omega(k_o+\kappa)-\omega(k_o))e^{i ...
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Show that $\|(pqp)^{-\frac{1}{2}}pq-pq\|<1$ for projections $p,q$ with $\|p-q\|<1$ in a $C^*$-Algebra

Let $p,q$ be projections in a (unital) $C^*$-Algebra with $\|p-q\|<1$. Definition of $(pqp)^{-\frac{1}{2}}$: Let $pAp$ be the $C^*$-Algebra of elements $pap$, $a\in A$ with unit $p$. Then $pqp$ is ...
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If $\forall v \in V, \ a(Tu,v)=(u,v)$ is $T$ a bounded an regular operator?

Let $V, H$ two Hilbert spaces infinite dimensional. If the bilinear form $a(.,.)$ satisfies There exists a constant $\alpha>0$ such that $\forall v \in V, \ a(v,v)\geq \alpha \|v\|^2.$ There ...
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35 views

If an operator has a cyclic vector, then its co-rank is at most $1$

Prove that if an operator on a Hilbert space has a cyclic vector, then its co-rank is at most $1$. My attempt: If an operator $T$ on Hilbert space $H$ has cyclic vector $u$, and $v \in$ ...
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A matrix has a cyclic vector iff every eigenvector is of geometric multilplicity 1

In a finite dimensional space, prove that a matrix has a cyclic vector iff every eigenvalue is of geometric multilplicity 1. I can show that if a matrix has a cyclic vector, then every eigenvalue is ...
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1answer
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Schrödinger operator with delta (zero range) interaction.

I am reading the book of Albeverio named Solvable models in quantum mechanics. In the first chapter it is explained how to realize the operator $"-\Delta+\delta_0"$ as a self adjoint operator on ...