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 decomposition of a an Operator on $\mathcal{L}(l^2(\mathbb{N})$

I have an exercise, where I do not really know how to solve it. Let $\{\lambda_n\}_n$ be a counting of the rational numbers in $[0,1]$ and define a bounded self-adjoint operator $T$ on ...
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Unique solution of $(L+\alpha I)z=y$

Let $L:X \rightarrow X$ be a bounded linear operator with bounded-inverse. How we can show that $(L+\alpha I) z= y $ has a unique solution for sufficiently small $\vert \alpha \vert$? If $Lx=y$, ...
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Power of positive operator

Let $H$ be a complex Hilbert space and $B(H)$ be the space of all bounded linear operator on $H$. Let $T\in B(H)$ be a positive operator ($\langle Tx,x\rangle\geq0$ for all $x\in H$) and $\alpha\in ...
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Bounded or unbounded operator?

Consider the operator $A\colon\ell^2\to\ell^2$ defined by $Au=(ku_k)$. Normally, this is not well-defined, since $(1/k)_{k=1}^\infty\in\ell^2$ but $A(1/k)=(1,1,\ldots)\notin\ell^2$; however, if one ...
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K-theory formulation of the index theorem

The Atiyah-Singer index theorem is one of the most important results in twentieth century's mathematics. It states that for an elliptic differential operator $D$ on a smooth, oriented compact manifold ...
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Construction of direct sum of Hilbert spaces

The following is a theorem of Takesaki's operator theory: I do not know why he puts $\bar H$ instead of $H$ for $n=-1,-2,\ldots\,{}$. Please help me. Thanks.
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Sot convergence of a net

The following are exercises of Conway's operator theory: I proved both exercises, but I confused about this point that in exercise 8, $T_i\to 0$ (sot), so based on exercise 6, $T_i^2 = T_i.T_i\to ...
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States: Liouvilleans

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider an invariant state: $\omega\circ\tau^t\equiv\omega$ Then the dynamics is unitarily implementable: ...
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Weak operator topology is the smallest topology on $B(H)$

Show that weak operator topology is the weakest locally convex topology on $B(H)$ such that every $\phi\in F(H)$ is continuous. (F(H) means finite rank operators on $H$). To show it , let $\tau$ ...
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Closed unit ball of $B(H)$ with wot topology is compact

The following is a Theorem of Conway's operator theory: I can not understand how he proves it. I think $\phi(\text{ ball B(H)})$ is compact if $\phi(\text{ ball B(H)})$ is closed subset of compact ...
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An inequality for positive operators

Let $S$ and $T$ be positive operators on a Hilbert space $\mathcal{H}$. Suppose that $S \le T$. Since the square root function is operator monotone, it follows that $S^{1/2} \le T^{1/2}$. Does the ...
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Existance of solution to integral Fredholm equation

I am a bit confused with the existence/non existence of a solution to the following equation: $$\int_{-T}^{T} R(t,\tau) x(\tau) d\tau = y(t), \forall t\in [-T,T]$$ where $y(t)=C$ (a constant ...
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1answer
103 views

On separable Hilbert space $H$, weak operator topology is metrizable on bounded parts of $B(H)$

The following is a theorem of Takesaki's operator theory: In this proof, weak topology means weak operator topology. I'm wonder why the theorem holds just for bounded parts of $B(H)$ and also ...
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1answer
131 views

Using lemma in proof

Hi please view the attachment. I am interested in how Lemma 1.11 is used in the proof of Theorem 2.10. Based on the statement of Lemma 1.11 it seems that in order to use Lemma 1.11 in we require ...
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1answer
28 views

Operator in Denominator

So I chanced upon this statement, and I'm not sure what is happening: $$ \left(c-\frac{1}{b}\frac{\partial}{\partial ...
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1answer
68 views

What is an adjoint operator?

The following conjeture is stated here: Every adjoint operator has a non-trivial closed invariant subspace. Reference 11 where adjoint is supposedly defined can be found here. But I don't have ...
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107 views

Is there a way to exploit local redundancy in a function to speed up Monte Carlo integration?

In every Monte Carlo method I've ever seen, $f$ must be recomputed from scratch for each point that is (somehow randomly) selected to contribute to the overall integral. However, most functions have ...
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Finite dimensional operator space is dense in trace class space

To show that $F(H)$ (the space of finite dimensional operators on a Hilbert space $H$) is dense in $L^1(H)$ (the space of trace class operators), suppose that $x\in L^1(H)$. Without loss of generality ...
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33 views

Trace class operator is an ideal

To show that trace class operators space is an ideal, we need to show that $\|uv\|_1\leq \|u\|\|v\|_1$ where $u,v \in B(H)$ and $\|u\|_1 = tr(|u|)$. Murphy in his book (C*-algebras and operator ...
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higher dimensional nature of tetrational forms

What is the exact value of fifth hyper operator of $3$ with height $1.1 $? I have for long time tried to figure out how to solve this but i still confuse,do anyone have a clue how to solve ...
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1answer
57 views

Differentiation operator is closed?

Got stuck in this problem: Let us denote by $X$ the linear space $C^1([0,1])$ equipped with the norm of $C^0([0,1])$ and consider the following statement: "The differentiation operator $L:X ...
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1answer
89 views

Blackwell's condition for a contraction: Why is boundedness neccessary?

I'm trying to understand the proof that certain operators $T$ are a contraction if they fulfill Blackwell's sufficient conditions. In particular, I try to understand why the operator $T$ has to map ...
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On the exponential form of a unitary matrix

A unitary matrix $U \in \large C_{n,n}$ can always be written in the exponential form: $U = e ^{iA}$ (1) where $A$ is Hermitian. My goal is to find the Hermitian matrix $A$, given the unitary ...
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1answer
98 views

A operator is unitary if and only if it is a surjective isometry

I'm trying to prove the following result. Let U be an operator of a Hilbert space H, then $U$ is an unitary operator $\iff$ $U$ is an isometry and $R_u = H$ ($U$ is onto and isometry) I tried to use ...
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66 views

Nonzero projection in an irredicible C*-algebra of minimal finite rank must have rank one

The following is a part of a theorem in Murphy's C*-algebras and operator theory: Let $A$ be a C*-algebra acting irreducibly on a Hilbert space $H$ and $q$ be a nonzero projection in $A$ of minimal ...
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37 views

$*-$ isomorphism between two compact spaces $K(H)$ and $K(H')$

The following is a theorem of Murphy's C*-algebras and operator theory: I think we can write the proof more easily than Murphy's. After show that $E'$ is an orthonormal basis for $H'$, define ...
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66 views

Sublinearity of Hardy-Littlewood Maximal Function on Sobolev Spaces

Define the (centered) Hardy-Littlewood maximal function by $$\mathcal{M}f(x)=\sup_{r>0}\dfrac{1}{m(B(x,r))}\int_{B(x,r)}\left|f(y)\right|dy,\ f\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{d})$$ We say ...
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1answer
24 views

Finding all operators that preserve a function

I'm not even sure what field of math this would be, and Googling "symmetry" and "functions" doesn't reveal what I'm looking for. Basically I want to find all $\{\hat{A}\}$ other than the identity ...
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1answer
32 views

strongly continuous mapping implies bounded mapping

Hi does anyone know how to show the result that if we have a relexive Banach space $X$ and a mapping $A: X \rightarrow X^{*}$ (not necessarily linear), which is strongly continuous, which means ...
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2answers
83 views

Definition of spectrum of an operator

The spectrum of a bounded operator is the eigenvalues for the corresponding matrix. Consider the following wiki link: http://en.wikipedia.org/wiki/Multiplication_operator In the example, it says ...
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weakly $p-$summable sequences in $L_{\infty}(\Omega, \mu)$

Let $1\leq p<\infty$ and $(\Omega, \mu)$ be a measure space and $f_{1},\ldots,f_{m}\in L_{\infty}(\Omega, \mu)$. What is the definition of weakly $p-$summable sequences $(f_{j})_{j=1}^m$ in ...
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55 views

Summary: Spectrum vs. Numerical Range

Reference A proof of the statement below is split into: Normal Operators: Spectrum vs. Numerical Range Spectral Measures: Spectrum vs. Numerical Range Problem Given a Hilbert space ...
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66 views

Check proof that operator in unbounded please

I have to show that $f:\mathcal{C}'[a,b]\rightarrow \mathbb{R}$ with $f(x)=x'(\frac{a+b}{2})$ is unbounded. Here $\mathcal{C}'[a,b]$ (the space of continuously differentiable functions) is to be ...
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Kreyszig's Functional Analysis Section 2.8: How is the canonical embedding map injective?

Let $X$ be a vector space over the field $K$ of the real or complex numbers. Let $X^*$ denote the vector space of all linear functionals defined on $X$, and let $X^{**}$ denote the vector space of all ...
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Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.8, Problem 4: How to show boundedness?

Let $f_1$, $f_2$ be the functionals defined on the normed space $C[a,b]$ of all continuous functions defined on the closed interval $[a,b]$ with the maximum norm be defined as follows: $$f_1(x) ...
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Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.8, Problem 3: What is the norm of this functional?

What is the norm of the linear functional $f$ defined on the normed space $C[a, b]$ of all functions defined and continuous on the closed interval $[a,b]$ with the norm defined as $$\Vert x \Vert ...
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Erwin Kreyszig Section 2.8, Problem 2: What is the norm of these two functionals?

Let $a$, $b$ be two real numbers such that $a<b$, and let $C[a,b]$ denote the normed space of all (real- or complex-valued) functions defined and continuous on the closed interval $[a,b]$ with the ...
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38 views

What's difference between spectrum and eigenvectors of an operator

Let $x$ be an operator in $B(H)$. By definition $\sigma(x)=\{\lambda \in \Bbb C ~; \lambda - x \neq inv \}$. Also to find eigenvalue of an operator we should find $\lambda$ such that $x\xi = \lambda ...
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weakly continuous linear map

The following is a Theorem of Murphy's C*-algebra and operator theory: To prove the theorem, the author claims compact linear map $u$ is weakly continuous. I know that every bounded linear map is ...
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Check proof about range of bounded linear operator.

I have to prove that the range $\mathcal{R}(T)$ of bounded linear operator $T:X\rightarrow Y$; $X,Y$ normed spaces need not be closed in $Y$. As a hint I'm given that I could consider ...
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207 views

An inequality concerning restricted isometry property

Let $A\in \mathbb{R}^{m\times n}$ be a matrix and let us denote by $A_S$ the submatrix of $A$ with the columns restricted to a set $S\subset [n]:=\{1,2,\cdots, \ n\}$. Then one says that the matrix ...
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1answer
36 views

Is this definition of a modulation operator ambiguous?

For $f \in L^2(\mathbb{R})$ and $b \in \mathbb{R}$, define a modulation operator $E_b$ from $L^2(\mathbb{R})$ to itself as: $E_b f(x) = e^{2\pi i b x}f(x)$ . Then the question is: for $a \in ...
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Is there a convention, law or axiom for associate operators when is a lack of brackets?

If a have an operator $\circledast:A\times A\rightarrow A$ and $a,b,c\in A$, then the expression $$a\circledast b\circledast c$$ Can be interpreted only as $(a\circledast b)\circledast c$ or is ...
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Image of a commutative C*-algebra

Let $A$ be an unital commutative C*-subalgebra of $B(H)$, and $\Omega$ be its character space. By spectral theorem $$\phi: B_\infty(\Omega)\to B(H);~~~~~f\to \int f \, dP$$ is a $*-$ homomorphism ...
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1answer
22 views

The Spectral Radius of a Product of Two Hilbert-Space Operators

I’m given a Hilbert space $ \mathcal{H} $ such that $ \dim(\mathcal{H}) > 1 $, and I’m supposed to construct two operators $ A $ and $ B $ on $ \mathcal{H} $ such that $ r(A B) \neq r(A) r(B) $. Is ...
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63 views

Nonunital C*-Algebras: Morphism Contractive

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. Consider a *-morphism $\pi:\mathcal{A}\to\mathcal{B}$. Then it is contractive: $\|\pi[\mathcal{A}]\|\leq\|A\|$ The proof I know ...
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1answer
39 views

Nonunital C*-Algebras: Closed Image

Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. Consider a morphism: $\pi:\mathcal{A}\to\mathcal{B}$. Then its image is closed: $\mathrm{im}\pi\subsetneq\overline{\mathrm{im}\pi}$ The proof I ...
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Construct a unitary operator U on H with prescribed spectrum

Given an infinite dimensional Hilbert space $H$. Let $|\lambda_k| = 1$ for $k = 1, ..., n$. Construct a unitary operator $U$ on $H$ such that $\sigma(U) = \{\lambda_k\}$ for $k=1,....,n.$ I can ...
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35 views

Construct a bounded linear operator S on H such that σ(S) = A

Given an infinite dimensional Hilbert space $H$. Let $A\subseteq \mathbb{C}$ be closed and bounded. Construct a bounded linear operator $S$ on $H$ such that $\sigma(S)=A$, where $\sigma(S)$ is the ...
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Extened of a representation

The following is a part of a theorem of Folland's book: Let $X$ be a compact space, $B(X)$ the space of bounded Borel measurable functions on $X$, and $C(X)$ the space of continuous function on $X$. ...