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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The set of all normal operators on a Hilbert space is not strongly closed

I need an example to show that the set of all normal operators on a Hilbert space is not strongly closed. Also I know that strong operator topology and strong* operator topology coincide in the set of ...
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1answer
22 views

Four-indexed infinite matrix

I've found on an ancient book a structure of the kind described below and I do not even know if it is a commonly known structure nowadays. For me it doesn't match neither a direct sum nor a direct ...
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1answer
71 views

Reiterate Volterra integral operator is a contraction

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 472 here) the statement that Volterra operator $A:L_2[a,b]\to L_2[a,b]$ defined ...
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1answer
27 views

$\ker (I-A)=\{0\}\Rightarrow\text{im }(I-A)=H$ for $A:H\to H$ compact

Let $T$ be the operator defined by $T:=I-A$ where $I:H\to H$ is the identity and $A:H\to H$ is a compact operator defined on Hilbert space $H$. In such a case, if we defined the chain of ...
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0answers
29 views

the spectral radius of normal operator

Let $H$ be a Hilbert space and $T$ be linear bounded operator in $H$. Prove that if $T$ is normal then the spectral radius of $T$, $$r(T)=\|T\|.$$ Is this TRUE?
4
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1answer
42 views

Green's operator for elliptic differential operator

Let $$ P:\Gamma(E)\rightarrow\Gamma(F) $$ be an elliptic partial differential operator, with index zero and closed image of codimension 1, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
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2answers
25 views

$P_M-P_N$ on which subspace of H is orthogonal projection?

Let M and N are two closed subspaces of Hilbert space H such that $N\subset M$. Also $P_M$ and $P_N$ are orthogonal projections on M and N respectively. It is clear that $P_M-P_N$ is again an ...
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0answers
25 views

C*-Algebras: Group vs. Derivation

Given a C*-algebra $\mathcal{A}$. Consider a *-derivation $\delta$. Does it always generate a group: $$\tau(t)=e^{it\delta}$$ But a group of *-automorphisms is a contraction group: ...
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2answers
30 views

A proof pertaining to the projector operator

Let $H_{1}$ be any subspace of a Hilbert space $H$, and let $H_{2} = H_{1}^{\bot}$ be the orthogonal complement of $H_{1}$, so that an arbitrary element $h \in H$ has a unique representation of the ...
2
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1answer
25 views

Toeplitz Operator is compact if and only if it has finite rank

A referee has pointed out to me that it is "well known that a Toeplitz operator is compact if and only if it has finite rank" and pointed me to R. Douglas: Banach algebra techniques in the ...
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15 views

When (weakly) compact operators have pre-adjoints?

Given a bounded linear operator $T\colon X^*\to X^*$ for some Banach space $X$. Then $T$ is an adjoint of an operator $S\colon X\to X$ if and only if $T$ is weakly* to weakly* compact. Are there some ...
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2answers
60 views

Dense subspaces, closed subspaces and unbounded operators in Hilbert spaces

Let $\mathcal{H}$ be a Hilbert space, and let $N\subseteq\mathcal{H}$. I found two interesting statements (without proof): if a closed subspace $N$ is such that $N^{\perp}=\{0\}$ (which is ...
0
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1answer
25 views

How to get the updating rules? after derivation

In the picture i brushed yellow, it dose make no sense to me to get formulas(2) and (3). If anyone could point out or give some references? Thanks a lot!
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2answers
22 views

Showing that $\mathcal{G}(\ell_2)$ is not dense in $\mathcal{B}(\ell_2)$ via the right shift

This is my question: Is $\mathcal{G}(\ell_2)$ is dense in $\mathcal{B}(\ell_2)$? I am attempting to show that it is not by showing that the right-shift - call it $T:\ell_2 \rightarrow \ell_2$ - ...
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0answers
7 views

Is it true that $\overline{\mbox{span}}\{z_1^n z_2^m H^2(T^2): |z_1|=|z_2|=1 \mbox{ and } m,n\in\mathbb{Z}_- \}=L^2(T^2)$?

Is it true that $\overline{\mbox{span}}\{z_1^n z_2^m H^2(T^2): |z_1|=|z_2|=1 \mbox{ and } m,n\in\mathbb{Z}_- \}=L^2(T^2)$? Here $H^2(T^2)$ is Hardy space on $T^2$.
2
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1answer
33 views

Lower bound of a positive self-adjoint operator

Let $A$ be a positive self-adjoint operator in a Hilbert space. Is there a way to show that $$\inf_{\|x\|=1}\|Ax\|=\inf_{\|x\|=1}|\langle x,Ax\rangle|$$ without using the spectral theorem and in a way ...
0
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1answer
17 views

A question on Compression spectrum

How to show that if $\lambda$ is in compression (or residual) spectrum of an operator $A$, then $\bar{\lambda}$ is an eigenvalue of $A^*$? Thanks.
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0answers
15 views

Relation between diferent definitions of Quasicontinuous functions

Defining the class of quasicontinuous functions by \begin{equation} QC=(H^{\infty }+C(\dot{{\mathbb R}}))\cap (\overline{H^{\infty }}+C(\dot{{\mathbb R}})). \end{equation} Where $H^{\infty }$ denotes ...
0
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1answer
34 views

Spectral Theory of an operator

If we define the spectrum of a bounded linear operator $T$ by $$\sigma(T)=\{\lambda\in \mathbb C:\ T-\lambda I \ \text{ has no inverse} \}.$$ What about $\sigma(T^{-1})$?
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36 views

sup norm of operator

Let $T$ be a compact linear operator defined as $$ T\circ u = \int_a^b k(x,y)\,u(y)\,dy, $$ where $k(x,y)\in C([a,b]\times[a,b])$ and $k(x,y)\ge0$ for all $x,y$, and $u\in C([a,b])$. Suppose that the ...
3
votes
2answers
48 views

Eigenvalue of a unilateral shift operator

Let $S:H\to H$ be a unilateral shift operator. I preferred in Example2.3.2 of Murphy's C*-algebras and operator theory that S has no eigenvalues. While $\{\lambda \in \Bbb C ; |\lambda|<1\} \subset ...
3
votes
1answer
30 views

What is “cyclic shift unitary” on $M_{n}(\mathbb{C})$?

Let $M_{n}(\mathbb{C})$ be the $n\times n$ complex matrices, and what is the "cyclic shift unitary of order $n$" on $M_{n}(\mathbb{C})$ ? (Maybe it is a very basic concept in functional analysis or ...
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0answers
21 views

Is the spectrum of a first order PDO always unbounded from both sides?

Let $E \to X$ be a smooth vector bundle over a compact Riemannian manifold $X$ and assume that $P:\Gamma(E) \to \Gamma(E)$ is a self-adjoint partial differential operator of order $1$. We think of ...
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1answer
27 views

Does Convergence of Maps Evaluated at Points Imply Convergence in Operator Norm?

Suppose that I have $T,T_n \in B_H$, for some Hilbert space $H$. Is the following implication true? $$ \|(T-T_n)x\| \rightarrow 0 \ \forall x\in H \ \Rightarrow \ \|T-T_n\| \rightarrow 0, \ \text{ie} ...
0
votes
1answer
45 views

Domain of a bounded linear operator on a Hilbert Space

In books, I always that a bounded linear operator on a Hilbert space is defined on all the Hilbert space, while an unbounded linear operator can not be defined on all the Hilbert space. Nevertheless, ...
0
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1answer
20 views

Showing self adjointness

$\pi:$ $Lx=\sum_{j=0}^{n}(p_{n-j}x^{(j)})^{(j)}$,$\,\,$ $x^{(j)}(a)=x^{(j)}(b)=0,\, j=0,1,...,n-1.$ where $p_{n-j}\in C^{n-j}[a,b]$ are real and $p_0(t)\neq0$ on $[a,b]$. I want to show that the ...
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2answers
36 views

Stone's Theorem and Functional Calculus

I've asked a few questions on here before regarding functional calculus but I am still having a bit of trouble. I have been reading up on Stone's theorem for unitary groups, and going through the ...
0
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1answer
31 views

an additive bijective map

Let H be a Hilbert space and $\Phi:B(H)\longrightarrow B(H)$ is an additive bijective map. If $\mathbb{R}I⊆\Phi(\mathbb{R}I)$, can we conclude by the bijectivity of $\Phi$ that ? ...
4
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1answer
34 views

Graph of weakly continuous linear operator

I have a few questions regarding the graph of an operator. Consider the operator $T:X \rightarrow Y$ between Banach spaces $X,Y$. Assume that $T$ is a linear operator which is (weak, weak)-continuous, ...
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1answer
17 views

Show $\sigma(T)=\sigma{(\overline{T^{*}})}$

Let $T \in B(H)$ be a bounded operator. Is $\sigma(T)=\sigma{(\overline{T^{*}})}$ true for $T$? $\textbf{TRY-}$ I have proved it is true for normal operator but could not do it for bounded ...
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1answer
13 views

Positive invertable element of a C*- algebra

The following is Theorem 2.2.5 of Murphy's C*-algebras and operator theory: Let $A$ be an unital C*-algebra and $a,b$ are positive invertable elements, if $a\leq b$, then $0\leq b^{-1}\leq a^{-1}$. ...
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0answers
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Green-Operator for Sturm-Liouville Differential equation compact on Sobolev space?

Let $g$ be Green's Function for a Sturm-Liouville differential equation. Is the operator $G: H_{0}^{1}(0,1) \rightarrow H_{0}^{1}(0,1)$ defined by $(Gf)(x) := \int_{0}^{1} g(x,y)f(y) dy, \quad f \in ...
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0answers
9 views

Visual notion of tangential gradient

Before I begin, this question is related to personal reading and is not in any way connected to an assessment/assignment. I am struggling to visualise the tangential gradient. As I understand it, the ...
0
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1answer
75 views

Is this equality true or it is not necessarily true?

Let $A$ and $B$ are two factor von neumann algebras that act on two infinite dimensional Hilbert spaces H and K respectively. Let $\Phi:A\longrightarrow B$ is an additive bijective map with some other ...
0
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1answer
29 views

Basis of $W^{1,p}_0\cap L^2$ using $(\lambda_i, v)_{H^s_0} = \mu_i(\lambda_i, v)_{L^2}$

Let $p > 1$. Define $\lambda_i$ by the eigenfunctions of the problem $$(\lambda_i, v)_{H^s_0} = \mu_i(\lambda_i, v)_{L^2}\quad\text{for all $v \in H^s_0(\Omega)$},$$ where $s$ is chosen ...
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3answers
31 views

Product of two positive compact, self adjoint operators

If we have two positive compact , self adjoint operators; $A$, $B$. Is the product $AB$ a positive operator?
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1answer
28 views

Spectral Measures: Subspace Characterization

Disclaimer This thread is related to: Spectral Measures: Subspace Decomposition It is meant to record. See: Answer own Question It is written as question. Have fun! :) Question Given a Hilbert ...
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2answers
28 views

solvability condition for differential operator

While reading the research article I came across following derivation, given a self-adjoint operator, \begin{eqnarray} L = \frac{d^2}{dx^2} + f(x) \end{eqnarray} \begin{eqnarray} L\psi_1(x) ...
0
votes
1answer
18 views

An example of an unbounded non-orthogonal projection in a Hilbert space

What is an example of an unbounded non-orthogonal projection in a Hilbert spaces? Does it exist? A non-orthogonal projection is an idempotent operator: $T^2=T$. So the question is: can such an ...
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0answers
13 views

Power series of bounded linear operators

If $f$ is a complex analytic function, one can define a matrix function $F$ using the Taylor series of $f$ by $$ F(A) = f(0) + f'(0)\cdot x + f''(0)\cdot \frac{A^2}{2!} + \cdots $$ If the radius of ...
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Laplace transform and Fourier transform of kernel

Suppose $p_t(x)=\frac1{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}$ is the Gaussian density. $p_t(x)$ is also the Green kernel of the heat equation in 1D: $(\partial_t-\frac12\Delta)u=0$. The Fourier transform ...
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1answer
39 views

Spectrum of a Self-Adjoint Operator is Real

Preparing for an exam in functional analysis, I'm trying to show that for a self-adjoint operator $A$, $\sigma(A) \subset \mathbb{R}$. I came across the following proof in the book (or rather, lecture ...
3
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1answer
32 views

Proving monotonicity of continuous linear functional

Hi I am interested in resolving the following problem from the bottom of page 147 from a paper I am revising: Given a function $$a: \Omega \times \mathbb{R} \times \mathbb{R}^{N} \rightarrow ...
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0answers
30 views

Find the eigenvalues and eigenvectors of T in V

Let $\mathbf{V}$ be the linear span of the functions 1, cos x, sin x. Let the operator T on V be given by the rule $T y(x)= y(x+\pi/4)$. Find the eigenvalues and eigenvectors of T in V. I'm not sure ...
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3answers
156 views

If a linear operator has an adjoint operator, it is bounded

This is a question I'm struggling with for a while: Let $H$ be a Hilber space. Let $T,S: H\rightarrow H$ be linear operators (not neccessarily bounded) such that for every $x,y\in H$: $\langle ...
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0answers
23 views

When is $\|\phi(|T|)\|=\|\phi(T)\|$ for $T\in B(H)$?

If $T\in B(H)$, $H$ a Hilbert space, then it has a polar decomposition $T=V|T|$, where $|T|=(T^*T)^{1/2}$ and and $V$ is a partial isometry. Let $\phi:B(H)\to B(H)$ be ucp (unital completely ...
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0answers
14 views

What conditions must an operator meet, to have only real eigenvalues?

Given the problem $Lu = \lambda u$, what properties must $L$ have, for all its eigenvalues to be real? An answer in the context of (partial) differential equations would be appreciated.
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0answers
18 views

Does any chaotic operator $T$ in infinite-dimensional Hilbert space is an isomorphism?

Suppose that any operator is an "Isomorphism" if it is mapping a hard equation to easier algebraic equations. Can I take this as a necessary criteria to judge that: "Any chaotic operator $T$ in ...
4
votes
1answer
54 views

$\sigma$-weak topology versus weak operator topology

The reference text for this question is: Pedersen, Analysis Now, GTM 118. The $\sigma$-weak topology on $B(H)$ (the bounded linear operators on a Hilbert space $H$) is the weak$^*$-topology on ...
2
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1answer
29 views

Polar decomposition

Every $x\in B(H)$ has a representation such as $x= u|x|$ (by polar decomposition) where $u$ is a partial isometry. In a paper, the author claims $$u = strong - \lim_{\epsilon\to 0} ...