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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12
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301 views

Is this a characterization of commutative $C^{*}$-algebras

Assume that $A$ is a $C^{*}$-algebra such that $\forall a,b \in A, ab=0 \iff ba=0$. Is $A$ necessarily a commutative algebra? In particular does "$\forall a,b \in A, ab=0 \iff ba=0$" ...
4
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1answer
93 views

Normal Operators: Meet

Given a Hilbert space $\mathcal{H}$. Normal Operators: $$\mathcal{N}(\mathcal{H}):=\{N:N^*N=NN^*\}$$ Borel Calculus: $$\mathcal{B}(N):=\{\eta({N}):\eta\in\mathcal{B}(\mathbb{C},\mathbb{C})\}$$ ...
0
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1answer
86 views

Normal Operators: “Zorn”

Problem Given a Hilbert space $\mathcal{H}$. Normal Operators: $$\mathcal{N}(\mathcal{H}):=\{N:N^*N=NN^*\}$$ (Possibly unbounded!) Borel Calculus: ...
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14 views

$A$ 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) ...
1
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1answer
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 $p_0,p_1,p_2\in C[a,b]$ ...
1
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1answer
40 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? ...
0
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1answer
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: ...
0
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1answer
513 views

Is my proof of characterisation of self-adjoint operators on complex Hilbert spaces okay?

I wish to show the following theorem: Let $T:H\to H$ be a bounded linear operator on a complex Hilbert space $H$. Then if $\left\langle Tx,x\right\rangle \in\mathbb{R}$ for all $x\in H$, then $T$ is ...
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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 ...
3
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2answers
34 views

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( ...
1
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0answers
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 ...
2
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1answer
40 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) ...
2
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1answer
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 ...
3
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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 ...
0
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40 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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17 views

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 ...
1
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1answer
22 views

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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64 views

show that a certain bounded linear operator does not exist

I'm trying to do problem 8 from section 5.10 on Evans' PDE book. Basically the problem asks if $U$ is a bounded open subset of $\mathbb R^n$ with $C^1$ boundary and $u \in L^p (U)$, show that ...
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118 views

Understanding dual spaces

My professor asked me to give him an example about dual spaces from our real life so I was thinking if we take the set of all our actions as the space X can the space of all our thoughts be the dual ...
2
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1answer
50 views

Why we need the measure of noncompactness

I just start studying the measure of noncompactness and I get confused. Why we try to measure the noncompactness of an operator? Is it to see if we can obtain a weak noncompactness?!
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17 views

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 ...
0
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1answer
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 ? ($C_c$ : ...
0
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1answer
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 ...
0
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1answer
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 ...
0
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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 ...
0
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1answer
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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1answer
20 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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1answer
40 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 ...
2
votes
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 $ ...
1
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1answer
26 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 ...
0
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2answers
27 views

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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17 views

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 ...
2
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0answers
48 views

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: ...
0
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1answer
68 views

Operator norm of symmetric Matrix in Hilbert Space with Hermitian Inner Product

Assume we have a postive definite real matrix $P$ and we define an inner product on a finite dimensional hilbert space $\langle x, y \rangle = x^\top P y$ and clearly the induced norm is $|| x || = ...
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2answers
35 views

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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2answers
68 views

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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2answers
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 ...
2
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2answers
61 views

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 ...
2
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2answers
45 views

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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11 views

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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2answers
43 views

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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0answers
40 views

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 ...
0
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1answer
37 views

Hardy space on the upper plane

Recently,I need to study something about Hardy space. However, many books only contain Hardy space on the unit disk. Is there any book having detailed description about Hardy space on the upper plane ...
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1answer
38 views

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 ...
3
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0answers
27 views

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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34 views

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 ...
0
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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 ...
0
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1answer
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 ...
3
votes
2answers
407 views

Spectral theorem of compact operators in Hilbert space

I am reading the following theorem from my lecture notes (English translation of German text). But I don't understand exactly what is meant from this theorem and the proof. Theorem. Let $H$ ...