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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If the bounded operators $X\to Y$ form a Banach space in the operator norm, is $Y$ necessarily Banach? [duplicate]

I have seen that if $Y$ is Banach, the set $B(X,Y)$ of bounded linear operators from $X$ to $Y$ is Banach in the operator norm. I was now wondering about the converse. Is it true? More precisely: ...
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Linear map from $L^1 \rightarrow L^{\infty}.$

I was wondering how I can show that any linear map $T: L^1(\Omega) \rightarrow L^{\infty}(\Omega)$ can be represented as an integral operator $$T(f)(x):=\int_{\Omega} K(x,y)f(y) dy.$$ Does anybody ...
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$A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras.

Is it true that $A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras, where $A^+$ denotes the positive elements in $A$. I read in Murphy 2.1.11 that this is true if $B$ is ...
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29 views

Proof of the continuity method, guidance

Let $\mathcal{B}$ be a Banach space, and $V$ a normed linear space. $L_0,L_1:\mathcal{B}\to V$ are bounded linear operators. Assume $\exists c$ such that $L_t := (1-t)L_0 + tL_1$ satisfies: ...
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48 views

Self-adjoint and positive operator minimal polynomial on complex inner product spaces

Suppose that T is a self-adjoint operator on the 2-dimensional complex inner product space. Suppose that the minimal polynomial of T is $$T^2-(a+c)T+(ac-|b|^2)I$$ a)Given that a, c are real numbers ...
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23 views

Positive operator minimal polynomial [duplicate]

Suppose that T is a self-adjoint operator on the 2-dimensional complex inner product space. Suppose that the minimal polynomial of T is $$T^2-(a+c)T+(ac-|b|^2)I$$ a)Given that a, c are real numbers ...
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38 views

Dominated convergence theorem for spectral measure

Okay, I posed my question maybe a little bit to vague: What I have in mind is the following: Let $L$ be a generator of a semigroup $(P_t)_{t \ge 0}$ with $\langle x,Lx \rangle \le 0$ defined on ...
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35 views

Question about means on linear maps from vector space of bounded sequences to $\mathbb{R}$

The definitions I am working with: $B$ is the vector space of bounded sequences $a=(a_n), n\in\mathbb{Z}$ for which there exists $C>0$ such that $|a_n|\leq C, \forall n$ 2. Mean on $B$ is a ...
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Matrix monotone operators Intuition

can anyone explain by intuition that a matrix(operator) $A$ is monotone? I know for normal functions if a matrix is monotone this means intuitively i can think of it as increasing, but hard to ...
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29 views

Von neumann contains the range projections of all of its elements

The following is a theorem of Murphy's C*-algebra and operator theory: I think it can prove easier, while I'm not sure about my proof : Let $a\in A$ be positive. Consider $C^*(a)$, and let ...
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37 views

Spectrum of unbounded operators

I am currently a little bit confused. I am aware of a theorem that says that any closed and densely defined operator satisfies $\sigma(T^*)=\overline{\sigma(T)}.$ On the other hand, the operator ...
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30 views

compact and normal operator is diagonalizable

In the following theorm, I do not know why $K^\perp = 0$ I just accept that $x = 0$ on $K^\perp$. For instance, $x = h\otimes h$ for $h\in H_{\|.\|=1}$ is a compact operator. Extend $\{h\}$ to a ...
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20 views

Power series coming from linear function in $\ell_3^*$

The Problem Suppose that $f\in \ell_3^*$. Show that the series $\sum_{n=1}^\infty f(e_n)^3$ converges. Discussion This problem is on a practice exam I have for a linear analysis course. My first ...
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65 views

Trace Class: Counterexample

This is a real question! Given a Hilbert space $\mathcal{H}$. Denote trace class by:* $$\mathcal{B}_\textrm{Tr}(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):\operatorname{Tr}|A|<\infty\}$$ Then ...
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221 views

How can I get eigenvalues of infinite dimensional linear operator?

What I want to prove is that for infinite dimensional vector space, $0$ is the only eigenvalue doesn't imply $T$ is nilpotent. But I am not sure how to find eigenvalues of infinite dimensional linear ...
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30 views

For what operators $A$ on a Hilbert space is the identity operator in the closure of the similarity orbit of $A$?

For a bounded linear operator $A$ on a separable Hilbert space, the similarity orbit of $A$ is the set $S(A)=\{WAW^{-1}: W \text{ is invertible}\}$. I am wondering that if the identity operator $I$ is ...
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21 views

Finite measure operator norm

Let $T^n: L^1(\mu) \rightarrow L^{\infty}(\mu)$ be a bounded operator for any $n$ and $\mu$ a probability measure. Is it then true that $||T^2||_{1 \rightarrow \infty} \le ||T||_{1 \rightarrow 2} ...
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48 views

Resolvent set/operator

Just a question here. Why do we study Resolvent operators and resolvent sets? Will there be any motivation or intuition behind this?
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101 views

Operator norm of positive operator.

I'm studying Reed and Simon's "Methods of Modern Mathematical Physics" Vol. 1 (http://www.math.bme.hu/~balint/oktatas/fun/notes/Reed_Simon_Vol1.pdf). In the proof of the square root lemma (p.196) they ...
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30 views

Definition of “Extension” of Bounded Linear Transformation

I have been given the problem of proving the B.L.T. Theorem for my homework which states, Every bounded linear transformation $\mathsf{T}$ from a normed vector space X to a complete, normed vector ...
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A restriction of a symmetric operator such that the range of (operator)+i is the same

I have this problem and I really can't see how to do it. Suppose that $C$ is a symmetric operator, $A\subset C$ and that $\operatorname{Ran}(C+i)=\operatorname{Ran}(A+i)$. Prove that $C=A$. ...
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22 views

Why is $\sqrt{T^*T}$ self-adjoint?

Let $T$ be a bounded linear operator over some Hilbert space $H$. Since $T^*T$ is a positive operator, it has a square root. Let $R=\sqrt{T^*T}$. Prove that $\forall u\in H, ||Ru||=||Tu||$. ...
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Existence of operator with certain properties on a Banach space

I ran across this question, and was a little puzzled by it. I neither know how to solve it, nor its meaning: Let $X$ be a Banach space, and let $A,B$ be bounded linear operators on $X$ such that $A$ ...
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27 views

Given an operator $ * $ and it's inverse $ \setminus $ when does $ x \setminus y = x * \left( 0 \setminus y \right) $?

Given a groupoid $ \left( M, * \right) $ with an neutral element $0$ and $\setminus$ being an inverse operator of $*$, what are the groupoid's properties for this predicate to be true? $$ \forall x, ...
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Trace Class: Relativeness

Given a Hilbert space $\mathcal{H}$. Consider a selfadjoint: $$H\in\mathcal{B}(\mathcal{H}):\quad H=H^*$$ Denote trace class: ...
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89 views

Spectrum of a closed operator

Could someone please explain this fact: if $A$ is a closed operator and $A^{-1}$ is a compact operator, then spectrum of $A$ consist only of eigenvalues? I forgot to mention that operator $A^{-1}$ is ...
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1answer
24 views

Conjugation: Boundedness [closed]

Given a Banach space $E$. Then for conjugations: $$C:E\to E:\quad C^2=1\implies\|C\|=1$$ How can I check this?
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72 views

What does $\operatorname{supp}(A)$ mean?

I'm looking at a paper (specifically this one). In the paper, we have a positive operator $A$, and the operator $\operatorname{supp}(A)$ is supposed to be a projection operator. Does anybody know ...
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25 views

What is the representation of the Grushin operator $G_\alpha$ in spherical coordinates $(r, \theta)$?

We known that the Laplace operator in spherical coordinates $(r, \theta)$ where $r = |x|$ and $\theta =\frac{x}{|x|} \in S^{N-1}$ is $$\Delta u = u''_{rr} + u'_r + r^{-2}\Delta_{S^{N-1}}$$ here ...
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Is the following differential operator closed (closabe)?

Let $L$ be the following differential operator. $L: C^2(\Bbb{R}^2_+)\to C^0(\Bbb{R}^2_+) $ $$Lf = \partial_x f(x,y) (y-x) + \partial_y f (x,y)(x-y) + \frac{1}{2} \bigg( \partial_{xx} f(x,y)x + ...
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transform between unitary operators

If I have a unitary operator $\exp(i\phi X)$, where $X$ is hermitian and $\phi\in\mathbb{R}$, is there a known way of finding an operator $\hat Y$ such that $$\exp(i\phi \hat X)=\exp(if(\phi) \hat ...
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Integration of $A$-valued functions (Functional Analysis)

Premise 1: my source is the Rudin - Functional Analysis. Premise 2: i'm not a mathmo so forgive for the mistakes A couple of question on the subject... An example of Banach Algebra is the set of ...
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C*-algebraic intrinsic definition for compactness of an operator?

Some properties of operators (normal, self adjoint, hermitian) have intrinsic definitions for any element of a $C^*$-algebra. Is there such definition for compact operators? Equivalently: Let ...
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1answer
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rad(T)=||T|| for non-normal T

It is well-known that for normal bounded operators $T$ on a Hilbert space one has $\mathrm{rad}(T)=\|T\|$ (where rad is the spectral radius). Are there any sufficient conditions under which a ...
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75 views

Eigenvalues of a certain product of matrices with special structure

Sorry for cross-posting from MO. Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the ...
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If $T$ is not bounded below then $T^*$ is not an open map

For $X, Y$ Banach spaces, and $T$ a bounded linear operator $T:X \to Y$, if $T$ isn't bounded below then $T^*:Y^* \to X^*$ isn't an open map. What methods can be used to prove something like this? ...
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operator ideals and their relationship to the geometry of Banach spaces, and other questions [closed]

Albrecht Pietsch wrote an excellent book on operator ideals in 1978, which I use frequently. But, I was reading the preface to his book, and I do not understand it. He writes (in English ...
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26 views

character space of a non unital abelian c*-algebra is locally compact

I would like to know why character space of a unital abelian c*-algebra is compact, while the charecter space of a non unital is locally compact. Why do we add 0 to character space of a non unital ...
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Vanishing of index of elliptic operators on odd dimensional manifolds

It is known that if $D$ is an elliptic differential operator on $M$ which is assumed to be odd dimensional, then index o $D$ vanishes. It essentially follows from the index formula in cohomology ...
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Convergence property of positive operator on $L^\infty$

Given a $\sigma$-additive measure space $(S,\Sigma,\mu)$ and a linear operator $$ U: L^\infty \to L^\infty $$ where $L^\infty$ is the space of essentially bounded measurable functions. Assume it is ...
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1answer
32 views

PDE and Linear operators

Consider a linear PDE which can be seen as L[u] = 0, where L is a linear differential operator on u. Is there any theory that tries to study the PDE by sutdying the kernel of L?
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Why do mathematicians say that “let an operator be represented by a matrix” instead of operator is the matrix?

For example, look at this sentence from Perko's text on dynamical system "It follows from Cauchy Schwarz inequality that if $T \in L(R^n)$ is represented by the matrix $A$ with respect to the ...
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Proposed proof Decomposition theorem

Let $\mathcal{A}$ be a unital Banach algebra. I want to prove that if $a \in \mathcal{A}$ and spectrum $\sigma(a) = \sigma_{1} \cup \sigma_{2}$ where $\sigma_{1} \cap \sigma_{2} = \emptyset$, ...
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Let $Y$ be a finite dimensional normed space, $X$ a normed space, and $T: X \to Y$ a surjective linear operator. Show that $T$ is an open mapping.

Let $Y$ be a finite dimensional normed space, $X$ a normed space, and $T: X \to Y$ a surjective linear operator. Show that $T$ is an open mapping. I think if I can show that $T(B_X)$ contains an ...
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1answer
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If a subspace of $X^*$ is weak*-dense, does it separate points?

Here $X$ is some normed space. I know the converse is true, but I don't know a proof for the other direction. That is, if $F\subset X^*$ is a subspace that is weak*-dense how would one show that ...
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Norm of derivative of rank one projector

Let $\phi(t)$ be a solution for the nonlinear Schroedinger equation \begin{equation} i\partial_t\phi(t)=-\Delta\phi(t)+(V*|\phi|^2)\phi(t) \end{equation} inside the Hilbert space $L^2(\mathbb{R}^d)$. ...
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64 views

Understanding how Nehari's problem connects with robust stabiliziation and Nevanlinna-Pick

I'm reading Young's "An Introduction to Hilbert space". In chapter 15 he writes about robust stabilization in control theory and ends with that this boils down to an interpolation problem called the ...
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Infinite dimension left shift operator over the complex vector space

Let $S$ be the left shift operator over the infinite complex vector field. Show that $$ \text{null}(S-I)^3=\text{span}\{(1,1,1,1,\ldots),(0,1,2,3,4,5,6,\ldots),(0,1,4,9,16,25,.....)\}. $$ To start I ...
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82 views

exercise, Hahn-banach theorem

I have this exerciose: Let $\Omega$ be a normed space. Prove that $\Omega$ is seperable if $\Omega^*$ is. It is in the chapter with the Hahn-Banach theorem, so I think I should use that ...
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2answers
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Prove that $A\geq I$ implies that $A$ is invertible.

Here's the question: Let $A$ be a positive operator on a (possibly infinite dimensional) Hilbert space. Let $I$ denote the identity operator. Suppose that $A \geq I$, which is to say that $A - ...