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 powers of contraction on Hilbert space converging to zero imply convergence of its adjoint to zero also?

In my functional analysis class I was met with the following problem: We suppose that $ \mathbb{H} $ is a Hilbert space and that T is a contraction operator on H (meaning $ ||T|| \leq 1 $ in the ...
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restrictions of closed linear operator to range of its powers

I am trying to prove that if $T$ is a closed densely defined operator on a Hilbert space(or Banach space), $\lambda$ is non-zero and $T_n$ is the restriction of T to range $R(T^n)$ for some n, then: ...
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13 views

Precedence of cross product and dot product

Which operator precedence is higher? The one of the cross product or the one of the dot product? Consider the following term: $$\overrightarrow {A}, \overrightarrow {B}, \overrightarrow {C}, ...
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A conjecture about traces of projections

Let $M_n$ denote the space of all $n\times n$ complex matrices. Define $\tau:M_n\rightarrow \mathbb{C}$ by $$\tau(X)=\frac{1}{n}\sum_{i=1}^n x_{ii},$$ where of course $X=[x_{ij}]\in M_n$. Recall that ...
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19 views

Closed operators

I was wondering whether the following statement is true or not? If $A$ is closed, then it follows from the closed-graph theorem that it is bounded iff $D(A)$ is closed. I found this in a chapter of ...
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$A$ and $A^*$ dissipative implies $D(A) \subset H$ is compact embedding

For selfstudy purpose I want to show the following: $H$ Hilbertspace, $D(A)$ dense subspace of $H$, $A\colon H \supset D(A) \to H$ linear closed dense defined operator. If $A$ and $A^*$ are both ...
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18 views

how is a compact embedding of infinite dimensional Banach spaces possible?

I'm looking at a dense defined closed operator $A\colon H \supset D(A) \to H$ with a Hilbertspace $H$ and $D(A)$ a dense subspace of $H$. In my notes there are some phrase like "if the embedding $D(A) ...
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23 views

An equivalent definition of self-adjoint operator?

Suppose the linear operator: $$\begin{array}{rcll} L:&C^2[a,b]&\longrightarrow& C[a,b]\\ &u&\longmapsto&Lu=p_0\ddot u+p_1\dot u+p_2u \end{array}$$ with $p_0,p_1,p_2\in C[a,b]$ ...
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15 views

$T+i\operatorname{Id}$ is an isomorphism for self-adjoint $T$

Let $T:H\to H$ be a self-adjoint continuous operator on a complex Hilbert space. Claim: $T+i\operatorname{Id}$ is an isomorphism and $\|(T+i\operatorname{Id})^{-1}\|\leq 1$. A few observations: ...
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A naïvely constructed extrapolation of a self-adjoint operator. Is it self-adjoint?

Let $\mathcal{H}$ be a real Hilbert space and let $A\colon D(A)\subset \mathcal{H}\to \mathcal{H}$ be an unbounded operator. Consider also a Hilbert triple $$ \mathcal{H}_+\subset \mathcal{H}\subset ...
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21 views

Differentiability of the norm in connexion with duality map

Let $(X,\|\cdot\|)$ be a Banach space and let $J$ be the duality mapping defined for all $x\in X$ by: $J(x)=\{x^∗∈X^∗\mid ⟨x^∗,x⟩=\|x\|^2=\|x^∗\|^2\}$, where $X^∗$ is the dual space of $X$. I'm ...
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1answer
38 views

How to prove this integral operator is bounded

Consider the integral operator $f\to g$: $$g(s)=\int_0^\infty\frac{f(t)}{t+s}\, dt$$ The above operator is the result of applying the Laplace transform twice. 1) What is the name of this operator? ...
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29 views

Operators of order $k$ between Sobolev spaces

It may sound like tautology, but I have a problem in proving that differential operator of order $k$ is an operator of order $k$. What exactly I'm asking for? Let $M$ be a manifold and $E,F$ be two ...
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34 views

Riesz Theorem on C[K], K compact

I'm studying Riez Theorem on Kreyszig's book: "Introductory functional analysis" , it states that "Let $l$ a bounded and linear functional on $C[a,b]$ (continuous functions on [a,b]) , then $l$ can ...
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1answer
39 views

A question regarding Eigenvalues

Note: $\psi,\psi^{\dagger} :\Bbb{R} \to \Bbb{C}$ and $x, \lambda_i , \hbar, m \in \Bbb{R}$ Say we know that $\lambda_1$ is a solution to the eigenvalue equation: $$\hat{\Pi}\psi(x)= \lambda_1 \psi(x) ...
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33 views

How to check if an operator is invertible?

Let $T_1 : C[a,b] \to C[a,b]$ be an operator defined by $$T_ v(x)=\int^b _a (x-t)v(t) dt$$ where $a \leq x \leq b$ and $v \in C[a.b]$ How can you check if the operator $T_1$ is invertible or not?
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calculating the abstract index of $C(T)$

Consider the following definition in operator theory: I'm reading an example of the abstract index of $\mathcal{A}$ in Zhu's An Introduction to Operator Albebras: Here $G_0(\mathcal{A})$ is the ...
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39 views

self-adjoint and orthonormal basis

Suppose $F=\mathbb{R}$. Let $A: V\to V$ (where $V$ is a finite dimensional inner product space over $F$) so that $A=A^*$ ("self-adjoint"), then there exists an orthonormal basis of eigenvectors and ...
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trace class and nuclear operators

Maurin (http://www.mscand.dk/article/viewFile/10641/8662) defines nuclear operators like this: A linear operator $A:\mathcal{H}_1\rightarrow \mathcal{H}_2$ where $\mathcal{H}_1$ and $\mathcal{H}_2$ ...
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spectrum of a positif operator !! [closed]

i have this question : we say that an $H$ operator is positive if we have : $<u,Hu>\geq0$ $\forall u\in D(H) $ so how to prove that for a self-adjoint operator $H$ we have : $H$ positive ...
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45 views

$S=\frac{-d^2}{dx^2}$ self-adjoint operator or not?

I have this simple question : In $L^2(]0,1[)$ let $S$ be the operator defined by : $D(S)=C_c^2(]0,1[)$ and $S=\frac{-d^2}{dx^2}$ is this operator self adjoint, and how to prove it ? ($C_c$ : ...
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36 views

Finding the norm of an operator

Consider the linear operator $T : C[0,1] \to \mathbb{R}$ defined by \begin{align*} T(x) := x(0) - \int_{0}^{1} x(t)\phantom{.}dt \end{align*} Show that $T$ is bounded and find its norm ...
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24 views

Using the definition of the operator norm

I am given the following problem: Using the definition $$\lVert L \rVert_{\text{op}}=\sup_{\vec{u} \in \mathbb{R}^d, \lVert \vec{u} \rVert=1}\lVert L\vec{u} \rVert$$ of the operator norm of a ...
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1answer
51 views

Commutating nilpotent operators

Is there good examples of collection of nilpotent operators that commute with themselves? Is there a good reference for a collection commutative nilpotent operators that commute with themselves or ...
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1answer
19 views

A counter example for adjoint of unbounded operators

I need a counter example for $(A+B)^*=A^*+B^*$, where $A$ and $B$ are unbounded operators on Hilbert space and $^*$ denotes the adjoint.
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Gauge invariance of a magnetic Schrödinger operator

Good morning, I am studying the properties of the magnetic Schrödinger operator $$ \mathcal{L}_A = \left( -\mathrm{i} \nabla -A \right)^2 = \left( -\mathrm{i} \nabla -A \right)^\dagger \left( ...
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1answer
36 views

Spectral density for the operator $A u = - u''$?

How to prove that the spectral density for the operator $A u = - u''$ on the whole real line is $$ e(x,y;\lambda) = \frac{\chi(\lambda) \, cos\lambda^{1/2} (x-y)}{2\pi \lambda^{1/2}} $$ where $\chi $ ...
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1answer
25 views

Sturm-Liouville operator with Dirichlet BC

I am trying to understand why Sturm-Liouville operator $$L(f)(x)=f''(x)-p(x)f(x)$$ with Dirichlet boundary conditions on $[a,b]$ is unbounded. $f$ is twice continuously differentiable, $p(x)>0$ is ...
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39 views

Boundedness and norm of a sequence operator

Let $s = \{s_{n}\}_{n=1}^{\infty}$ be a fixed and bounded sequence of real numbers, i.e. $s \in (\ell^{\infty},\|\cdot\|_{\infty})$. Consider the operator $T_{s} : \ell^{2} \to \ell^{2}$ defined ...
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57 views

Boundedness and norm of a linear operator

Consider the linear operator $T : C[-\pi,\pi] \to \mathbb{R}$ defined by $$ Tf := \int_{-\pi}^{\pi} f(t)\sin(t)\phantom{.}dt $$ Show that $T$ is bounded and find its norm $\|T\|$. Consider ...
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Generalization of matrix inversion lemma

I am looking for an operator version of matrix inversion lemma. To be specific, does the identity also hold for operators defined on general (infinitely dimensional) Hilbert space, possibly with ...
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Meaning of non-degenerate representation in $C^*$-algebras

A representation of a $C^*$-algebra, $A$, is a pair $(H,\pi)$ where $H$ is a Hilbert space and $\pi$ is a *-homomorphism from $A$ to $B(H)$. A representation is non-degenerate if $\{\pi(a)h:a\in A, ...
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open sets in a Banach space are locally connected

I'm reading a proof of the following theorem in operator algebra and I don't understand the first sentence: Would anybody show me why the following statement is true? Let $X$ be a connected ...
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26 views

Exponent of an Exponential Operator

There is a problem in my textbook that asks me to prove the following: For a bounded operator $A$ on a Hilbert space, prove that: $$(e^A)^n = e^{An} $$ for any natural number, $n$. However upon ...
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What is this operator topology?

Let $X$ be a separable Banach space with (norm $1$) Schauder-Basis $\{e_n\}_{n\in\mathbb N}$. Denote for $x\in X$ with $|\cdot|_x$ the seminorm on $\mathcal L(X)$ given by $|A|_x = \|A x\|$. Consider: ...
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Invariant subspaces for this linear extension of operators

Let $(e_k)$ be a total orthonormal sequence in a separable Hilbert space $H$ and let $ T: H\to H$ be defined at $e_k$ by $T(e_k)=e_{k+1}$ , $(k=1,2,\cdots)$ and then linearly and ...
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Determining eigenvalues of a differential or integral operator in Matlab?

Say I have a differential operator such as $L[\phi] = \frac{\partial \phi}{\partial x}$, or $L = \Delta \phi$, or an integral operator such as $L[\phi](x) = \int_{\partial D} \log(x - y) \phi(y) ...
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Spectrum of the derivative operator: What's wrong in my argument?

Consider the Banach space $X=C[0,1]$ of continuous functions $f:[0,1]\to\mathbb{R}$ equipped with the supremum norm. If we consider the following unbounded operator $A$ defined on its domain ...
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1answer
87 views

Proving an isometric dilation of a non unitary operator on Hilbert space implies infinite dimensional space involving matrices

I have been given this exercise in my Operator theory class dealing with operators on Hilbert spaces, which reads as follows: Let H be a Hilbert space. We are to prove, in two distinct ways, that ...
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Density of sets whose image is dense.

This is probably easy, but I can't think of an answer. Assume $X$ is a Banach space and $A$ is a (not assumed closed) subspace of $X$. Let $T:X \to X$ be a bounded operator, which is also injective. ...
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Spectral Measures: Integrability

I really need this as tool for other threads! Given a Hilbert space $\mathcal{H}$. Also a Borel space $\Omega$. Consider a spectral measure: $$E:\mathcal{B}(\Omega)\to\mathcal{P}(\mathcal{H}):\quad ...
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Eigenvectors Operators and Unilateral Shifts

We showed that a (non-zero) compact self-adjoint operator on a Hilbert space always has an eigenvector. Let $V:l^2(\mathbb{N})\to l^2(\mathbb{N})$ be the unilateral shift, the unique bounded operator ...
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Unitary Operators & Compact Self-Adjoint Operators

Let $U$ be a bounded operator on a Hilbert space. Show that the following are equivalent: I. $U$ is surjective and $\|Uv\|=\|v\|$ for all $v\in H$; II. $U$ is surjective and $\langle ...
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Bounded Operators: Topological Dual

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider the bounded operators: $$\mathcal{B}(\mathcal{H},\mathcal{K}):=\{T:\mathcal{H}\to\mathcal{K}:\|T\|<\infty\}$$ Regard the linear ...
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Bochner Integral of Positive Operators

So we have two function spaces (real or complex) X and Y (think $L^p$) and we say that a linear operator $P : X \to Y$ is positive if $f \geq 0$ implies $P(f) \geq 0$. I'm curious when a general ...
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1answer
21 views

why is the order of operations (for multiplications and division) giving different result?

Firstly sorry if this is tagged incorrectly or blindly obvious but it is confusing me a lot and I am not sure what category it would fall under. I have a particularly scenario where I am using the ...
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16 views

What is “analytic vector for closed operator”?

I need the defenition of "analytic vector of closed operator that acts on Hilbert space". I cant find it in google and in my textbooks (Khelemsky "Lectures And Exercises on Functional Analysis"), I ...
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Question on operator theory classes of operators on Hilbert spaces

I was recently tackled by this in my class on operator theory dealing with operators on Hilbert spaces: We are to find and prove the inclusion relations between the classes of operators: ...
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12 views

Boundedness of a dirichlet form

Suppose $\mu$ is a finite measure on some space $\Omega$ (can simply be the Lebesgue measure on $[0, 1]$ or something like it). Let $S_1$, $S_2$ be two densely defined, symmetric, nonnegative ...
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83 views

In which sense is composition a tensor product

Let $\Phi\colon U\to V$ and $\Psi\colon V \to W$ be linear operators, and consider their composition $$ \Psi\circ \Phi $$ The operation, $$\circ:\mathcal{L}(U,V)\times\mathcal{L}(V,W)\to ...