Tagged Questions

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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Restriction of a spectral measure

Let $x$ be a self-adjoint operator on $H$. By spectral theorem, there is a spectral measure $\mu$ correspondence to $*-$ homomorphism $\pi:C(\sigma(x)) \to B(H)$ such that $x=\int_{-||x||}^{||x||} ...
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Specific Type of Dominated Convergence (Spectral Measures)

Reference See Birman and Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, chapter 5 subparagraph 4.1, page 133... Question It is introduced a specific type of convergence, ...
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10 views

Correspondence between bounded sesquilinear forms and bounded linear operators

Let $H,K$ are Hilbert spaces, I want to show there is an isometric linear correspondence between bounded sesquilinear forms $S(H,K)$ and bounded linear operators $B(H,K)$. ( $\Phi: B(H,K)\to S(H,K)$ ...
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1answer
19 views

Positivelinear operator on $L^p$-spaces

Suppose $1<p<\infty$. A linear operator $T \colon L^p(\Omega)\to L^p(\Omega)$ is positive if $f \geq 0$ imply $T(f)\geq 0$ (where $\Omega$ is a measure space). 1) Does there exist a positive ...
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1answer
30 views

Spectral Measures: Square Root Lemma

Given a Hilbert space $\mathcal{H}$. Consider a densely defined closed operator $A:\mathcal{D}(A)\to\mathcal{H}$. This gives rise to operators: $$A^*A:\mathcal{D}(A^*A)\to\mathcal{H}$$ ...
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1answer
40 views

A question about sublinear functionals

Could you please give me hints may leads to prove the following: Let $X$ be a real vector space, $\,p_1,p_2:X\to\mathbb R\,$ be two sublinear functionals, and $\,f:X\to\mathbb R\,$ be a linear ...
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2answers
52 views

About the adjoint operator and weak operator topology.

Let $X,Y$ be Banach spaces. Let $\lbrace{S_n\rbrace}\subset\mathcal{L}(X,Y)$, and $T\in\mathcal{L}(X,Y)$, such that $S_n\xrightarrow[n\to\infty]{WOT}T$, that is: $$\langle ...
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Bounded-below multiplication operator on Hardy space

Let $H^2(\Delta^2)$ denotes the Hardy space on the bi-disc $\Delta^2$ and $M_f :H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be multiplication operator by $f\in H^\infty(\Delta^2)$ defined by ...
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3answers
35 views

Show that an isometric linear operator $T:H\to H$ satisfies $T^* T=I$, where $I$ is the identity operator on $H$.

Show that an isometric linear operator $T:H\to H$ satisfies $T^* T=I$, where $I$ is the identity operator on $H$. I've been stuck on this for a while and don't really know where to start.
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1answer
36 views

Showing an operator is essentially self-adjoint

I have a question about checking if an operator is essentially self-adjoint. Given the operator $$H=-\frac{1}{2}\partial^2_{r}-\frac{1}{r}\partial_r$$ with domain $C^{\infty}_0((0,\infty))$ (i.e. ...
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24 views

Boundedness Spectral Triple Axioms for de Rham Complex

In Connes' axioms for a spectral triple $(A,H,D)$, they have a representation of an algebra $A$ in bounded operators on a Hilbert space $H$, and (unbounded) operator $D$, such that $[D,a]$ is bounded. ...
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38 views

What's the difference between a costant $k$ and $arity = 0$?

What kind of mathematical entities can have $arity = 0$ without being a constant ? Or there is a concept that generalize the concept of constant and I can't see it ? Background: I am having some ...
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15 views

Multiplication operator is bijective?

Let $H^2(\Delta^2)$ denote the Hardy space on the polydics and $M_f:H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be a multiplication operator by $f\in H^\infty(\Delta^2)$. Is the operator is bijective? ...
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1answer
48 views

Measurable functional calculus

I am struggeling with this exercise: Let $T \in L(H)$ be a self-adjoint operator and $\Psi$ be a measurable (Borel) functional calculus on the spectrum of $T$. For a Borel set $\Delta \subset \sigma ...
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38 views

Is there a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ on $L^{2}[0,2\pi]$ with periodic conditions?

Let $\mathcal{H}=L^{2}[0,2\pi]$, and let $L=-\frac{d^{2}}{dx^{2}}$ on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions $f$ on $[0,2\pi]$ with $f''\in\mathcal{H}$ and ...
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1answer
36 views

How to show $e^{-x}$ is a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ in $L^{2}[0,\infty)$?

Let $\mathcal{H}=L^{2}[0,\infty)$. How can one easily show that $e^{-x}$ is a cyclic vector under the $C^{\star}$ subalgebra of operators on $\mathcal{L}(H)$ generated by all resolvents $(L-\lambda ...
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2answers
34 views

Conditions for a given operator being compact.

So I was given this question in class, and I thought it looked easy enough at first glance, but actually trying to do it, I have gotten quite stuck on the "only if" part. Let $T\colon \ell_\infty ...
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3answers
43 views

extending a bounded linear operator

So I have a homework question which I have no idea how to start. Let $E_0$ be a dense subspace of the normed space $E$. Let $T_0:E_0 \rightarrow F$ be a bounded linear operator into the Banach space ...
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14 views

What does the Stein–Weiss Interpolation Theorem say?

I was looking for the statement for the Stein–Weiss Interpolation Theorem, but I cant find it anywhere on internet.
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0answers
15 views

What is iterative integral operator?

I don't know what is iterated integral operator, and why $K^{(3)}$ is square-integrable implies $K$ is compact. Can anyone help?
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1answer
9 views

Showing T intertwines $D_T$ and $D_{T^*}$ using Spectral Theorem

Suppose $T$ is a contraction on a Hilbert space $H$ (separable, if you wish). $D_T=(I-T^*T)^{1/2}$ and $D_{T^*}=(I-TT^*)^{1/2}$. I want to show that $TD_T=D_{T^*}T$. I had done this before using a ...
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1answer
68 views

T is not compact operator

I want to show that if $T$ is a bounded operator between two Hilbert spaces and $T$ is not compact then there exists an orthonormal sequence $y_{n}$ and an $R>0$ such that $\forall n\in ...
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0answers
29 views

Self-adjoint operator and eigenbasis

Let us assume that we have a self-adjoint operator $A: D(A) \subset L^2 \rightarrow L^2$ and we know that $A$ has a purely discrete spectrum and the eigenvalues of $A$ are simple. Does that mean that ...
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60 views

Derivatives of Differential operators

Let $V=H^1(\Omega)$ and $U = L^{\infty}(\Omega)^2$. Take for example, a differential operator, $L:U\times V \rightarrow V^* $ such that: \begin{equation} L(u,y) = \Delta y + \vec{u}(x)\nabla y ...
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22 views

Equivalent definitions of the trace of a Hilbert-Schmidt operator

I am currently reading the book Spectral Methods in Automorphic Forms, and Iwaniec defines the trace operator in a different way than I am accustomed to. Throughout, assume that everything converges ...
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1answer
75 views

How to show a Borel Operator Measure dilates to a Spectral Measure?

Does anyone know a simple proof of the following theorem stating that a positive Borel operator measure $P$ on $\mathbb{R}$ can be written as $V^{\star}EV$ for a Borel spectral measure $E$? ...
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85 views

Composition of projections has a fixed point in a Hilbert space

Let Let H be a Hilbert space with an inner product ⟨⋅,⋅⟩ : H×H→R, and induced norm $∥⋅∥ : H→R_+$ Let $C_1$ and $C_2$ be closed, convex, nonempty, disjoint subsets of $H$ with at least one of ...
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1answer
59 views

Spectral Measures: Spectral Subspaces

Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. Consider a normal operator $N:\mathcal{D}\to\mathcal{H}$. Denote its associated Borel spectral measure by: ...
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79 views

Closure in a Hilbertspace

Define for a self-adjoint pure contraction $S$ (remember: $\|S\|\leq1$ and $\pm1\notin\sigma_p(S))$ on a Hilbert space $\mathcal{H}$ the following set: $C_c^*(S):=\{g(S):g\in C_c(\hat{\sigma}(S))\}$ ...
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1answer
24 views

Reducing Subspaces: Nonexample?

Given a Hilbert space $\mathcal{H}$. Consider an operator $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose there exists a closed subspace $Z\leq\mathcal{H}$: $$TZ\subseteq Z,TZ^\perp\subseteq Z^\perp$$ ...
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1answer
30 views

a question about contractions on Hilbert spaces

Let $\cal{H}$ be a Hilbert space, $T_1,T_2\in\cal{B(H)}$, $\|T_1(h_1)+T_2(h_2)\|^2\leq\|h_1\|^2+\|h_2\|^2$ for all $h_1,h_2\in\cal{H}$. $T_1T^\ast_1+T_2T^\ast_2\leq I$. Then can we verify that 1 ...
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1answer
46 views

Finding the norm of integral operator

I have the following operator: $A: (C^1[0;1];|||\cdot|||)\rightarrow(C^1[0;1];||\cdot||_{\infty})$ $Af(x)=\int_0^x f(t)dt$ where $|||f|||= ||f||_\infty+||Af||_\infty$ What is $||A||?$ I wrote ...
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1answer
18 views

Isometric Operators: Common Core

Given a Hilbert or Banach space $\mathcal{H}$. Consider two closed operators $S:\mathcal{D}(S)\to\mathcal{H}$ and $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose they're isometric on a common core ...
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1answer
77 views

$\langle Tx,x\rangle =0$ , then T is zero

I just wanted to be sure about something. The implication $\langle Tx,x\rangle =0$ , then T is zero , holds only if $T$ is self-adjoint right? If $T$ is an arbitrary operator, we need to have $\langle ...
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1answer
48 views

The set of analytic functions on unit circle is not a C*-algebra

Let $\mathbb{D}$ be the open unit disc on the complex plane and consider the set $$A=\{f\in C({\rm cl}\, {\Bbb D})\colon f \text{ is an analytic function on } {\Bbb D}\}.$$ It is certainly closed ...
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1answer
18 views

Is $C(\Omega)$ a C*-algebra if $\Omega$ is not locally compact, nor compact?

We always say if $\Omega$ is compact or locally compact, then C(\Omega) is a C*-algebra. Now is $C(\Omega)$ a C*-algebra if $\Omega$ is not compact nor locally compact? If not, I want to know which ...
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1answer
26 views

spectral projection of an element in a C*-algebra

I'm studying Takesaki's Operator theory and I preferred "spectral projection "in the page 43 of this book while he didn't speak about it before. I searched it, but I could not find it. Please explain ...
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22 views

Getting U.C.P map on group operator algebras using Fell's absorbtion principle.

I'm struggling a bit with this theorem: Let $\Gamma$ be a discrete group and $\mathbb{C}\Gamma$ be the group ring of $\Gamma$ i.e. the set of formal sums $\sum_{t \in \Gamma} \alpha_t t$. Furthermore ...
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1answer
16 views

Characterization of multipliers on the Dirichlet space of holomorphic functions

Can someone tell me whether there is a characterization of boundedness of multiplication operators on the Dirichlet space of the unit disc? These are holomorphic functions $f$ on the unit disk $D$ ...
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0answers
24 views

Operator for scaling a function?

Let $\mathbb{F}$ denote the set of functions of the form $f: \mathbb{R} \to \mathbb{R}$. I am interested to know whether there exists a well-known linear map $T_\alpha: \mathbb{F} \to \mathbb{F}$ ...
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3answers
81 views

Selfadjointness of the Dirac operator on the infinite-dimensional Hilbert space

I am a physicist, so my background in functional analysis is limited only to basics. However, I would like to prove that the free Dirac operator is selfadjoint (or Hermitian, or neither). The free ...
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1answer
23 views

norm of inverse of a bounded operator [duplicate]

Are there any conditions in which norm of inverse of a bounded operator T is equal to reciprocal of norm of the operator T.
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1answer
47 views

boundary conditions for operator

if you have a Schrödinger operator on a sphere ( $\mathbb{S}^2$) $-\Delta_{\theta,\phi} \psi(\theta,\phi) + V(\theta) \psi(\theta,\phi) = E\psi(\theta,\phi),$ where the potential does not depend on ...
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1answer
31 views

Positive linear functional on an involutive Banach algebra

Why is every positive linear functional on an involutive Banach algebra with a bounded approximate continuous?
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1answer
27 views

Approximate unit of an involutive Banach algebra

I know that every C*-algebra has an approximate unit. I have two questions: why we cannot show that every involutive Banach algebra has an approximate unit? I need an example of an involutive Banach ...
2
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1answer
26 views

The spectrum of $L:=-\Delta+V(x)$ on complex $L^2(\mathbb{R}^N)$ and real $L^2(\mathbb{R}^N)$

In general, when one talks about the spectrum of an self-adjoint operator, it is naturally considered in a complex Hilbert space (say $L^2(\mathbb{R}^N,\mathbb{C})$). Moreover, the spectral ...
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1answer
25 views

Positive elements in a Banach algebra

Let $A$ be a unital Banach algebra. If $a$ is an element of $A$ with $||1-a||_{sp}<1$, then there exists $b\in A$ such that $b^2=a$. Furthermore, if $A$ is an involutive Banach algebra and if $a$ ...
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35 views

On calculating spectral projections

Consider following operator from this paper; Let $h$ be any function in $L^1$ relative to the measure $g(w)dw$ and $K\in\mathbb{C}$ Consider the linear operator $B$ on $L^1$ defined by $$(Bh)(x) = ...
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0answers
16 views

Eigenvalues of a Self-Adjoint Operator

It is easy to see that eigenvectors corresponding to distinct eigenvalues of a self-adjoint operator $T$ are mutually orthogonal. However, from this, it is supposed to be easy to see that for a given ...
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0answers
32 views

Laplacian on $\mathbb{S}^2$ has a pure point spectrum

Consider an operator $T = -\Delta + V(\theta)$ where $V(\theta)$ is $C^{\infty}$ and $T : C^{\infty}(\mathbb{S}^2) \subset L^2(\mathbb{S}^2)\rightarrow C^{\infty}(\mathbb{S}^2).$ I was wondering why ...