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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Power series expansion of an Operator.

I've been reading a paper called "Separation of variables for the quantum $Sl(2,R)$ spin chain" in which the author at one point does a power series expansion I do not understand. The problem is this ...
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63 views

Extending finite rank operators

Suppose $Y$ is a closed subspace of Banach space $X$ and $T:Y\to X$ is a bounded finite rank operator. Can we extend $T$ to $\tilde{T}:X\to X$, in the sense that: $T=\tilde{T}$ on $Y$ ...
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Find the iverse of the followning bounded operator?

The following definition and Theorem are given in the book "A short course on operator semigroup" by the author "K-J Engel and R Nagel". Sectoral operator: A closed linear operator $(A,D(A))$ in ...
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89 views

Showing the compactness of a limit operator.

I was trying to solve this exercise from Kreyszig's book, section 8.1 exercise number 10. My attempt was try to show that the operators in the sequence are bounded, but I don't find it. If this fact ...
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47 views

Dense domain of Unbounded Operator

Let $H$ a Hilbert space and $A:D(A)\subsetneq H\rightarrow H$ a dissipative, unbounded linear operator with $R(A)=Im(A)$ closed in $H$, such that exist $A^{-1}$, bounded linear operator. How I can ...
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44 views

Show for the Hamilton's operator $H$ that $\overline{(H, C_0^{\infty}(\mathbb{R}))} = (H, W_2^2(\mathbb{R}))$ using Fourier transform

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable real-valued function defined on $\mathbb{R}$ bounded with its first derivative. Consider the Hamilton's operator $H$ such that: ...
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2answers
36 views

Hahn Banach Theorem: Clarification on meaning of extending a functional?

Hahn Banach Theorem: Given linear (vector) space $\mathbb{X}$, define $u \in \mathbb{L} \subset \mathbb{X}$, $A,B,C$ functionals, A sublinear. $A:\mathbb{L} \to \mathbb{R}, B:\mathbb{L} \to ...
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56 views

Taylor expansion of $\frac{1}{1-D}$, where $D$ is the differential operator

I understand that we can represent $e^D$ simply as a power series of D. But what about functions of D which are not entire on the complex plane? What if the function has no taylor expansion, or if it ...
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1answer
31 views

Partial Isometries: Final

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$J\in\mathcal{B}(\mathcal{H},\mathcal{K}):\quad P:=J^*J$$ By the C*-property: $$J=JJ^*J\iff P^2=P=P^*$$ Note that in any ...
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Construct an operator that fixes the equivalence class of Cauchy sequences

Let $X$ be a Banach space and $\overline{X}$ be its unique completion. We know that $\overline{X}$ can be partitioned into equivalence classes of Cauchy sequences via the relation $\sim$: $$ \{x_n\} ...
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39 views

(Operator) norm inequality for continuous functions

Let $f,g$ be two non-negative continuous functions on $[0,\infty)$ satisfying $f(t)g(t)=t,$ $\forall t\in[0,\infty)$. Let be $A$ be a bounded linear operator acting on a Hilbert space. Then I was ...
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Are all matrices linear operators?

Given $A \in \mathbb{K}^{n\times m}$ a matrix, can we think of $A$ as an operator? In what context do matrices satisfy the definition of operator?
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95 views

Is there a $C^{*}$ algebra with these properties

Is there a unital C* algebra A which is NOT simple but satisfies the following two conditions: 1)A has trivial center 2)A has a faithful trace such that every zero trace element lies in the closure ...
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25 views

A question on Operator of a Banach Space

For any $x \in X$ where $X$ is a Banach space, is there a non-trivial bounded operator $T \in B(X)$ such that $T(x)=x$? I mean is there any way to verify the existence of such an operator for any $x ...
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66 views

Solution to Equation $Ax=f$ in Hilbert Space

Question. Let $H$ be a separable Hilbert space with complete orthonormal basis $\left\{u_{k}\right\}_{k=1}^{\infty}$, let $H_{n}:=\text{span}\left\{u_{1},\ldots,u_{n}\right\}$, and let ...
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2answers
61 views

Definition of associative algebra over a field

In the definition of an algebra over a field in the wiki entry , it states that an algebra over a field is a vector space equipped with a bilinear product. Question: Does anyone know how a bilinear ...
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Absolutely Continuous Spectrum and Norm of Resolvent

Problem. Let $H$ be a Hilbert space, and let $A:H\rightarrow H$ be a bounded, linear operator. Suppose $A$ has purely absolutely continuous spectrum and $\sigma_{ac}(A)=[0,1]$. Find the set of ...
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Is the following inequality true : $(||f(|A|)||)(||g(|A^*|)||) \geq ||A||$?

Please help me on this. Give an example of an operator $A\in B(H)$, such that $$(||f(|A|)||)(||g(|A^*|)||) < ||A||$$ where $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ ...
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75 views

Weighted shift operator is Hilbert-Schmidt

If $W : \ell^2 \to \ell^2$ is the weighted shift operator defined by $$W(x_1,x_2,x_3,\ldots)=(0,x_1,\frac 12x_2,\frac 13x_3,\ldots),$$ how can I show that $W$ is Hilbert-Schmidt? If I have ...
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36 views

Operator norm and equivalent definitions

From the definition of the operator norm, we have: $||T||_{op}=\inf\{c\in \mathbb{R}^+:||Tv||\leq c||v||, v \in V\}$ If $T: V \rightarrow W$ is a linear map between two normed vector spaces. I have ...
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2answers
38 views

Adjoint of $\lambda I - T$

Given a selfadjoint (maybe unbounded) operator $T$ on a Hilbert space $H$, I want to calculate the adjoint of $\lambda I - T$ for a $\lambda \in \Bbb C$. I am tempted to argue as follows: ...
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1answer
50 views

Injectivity of an operator when it is extended

Suppose X be a Banach space, T be a bounded operator on X and Y be a T-invariant subspace of X (not necessary closed subspacace). If T is injective on Y, can we say T will be also injective on closure ...
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Square root of Differential Operator Self-Adjoint

If $H=L^2(0,2)$ $Aw=w^{(4)}, D(A)=H^4(0,2)\cap H^2_0(0,2)$ and $Bv=-(d(x)v')', D(B)=\{v\in H^1_0(0,2)\mid dv' \in H^1(0,2)\}$ where $$d(x)=\begin{cases}1\text{ if }0<x<1 \\ 2\text{ if ...
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73 views

can i prove or reject it?

Let $\Phi$ be a surjective map on an algebra $\mathcal{A}$ which satisfies the following condition for a fixed arbitrary $\epsilon \in \mathbb{C}$ which $\epsilon\neq1,-1$ and for fixed ...
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1answer
66 views

Contraction: Precompactness

Given Banach spaces $X$ and $Y$. For precompactness: $$\tau\in\mathcal{C}(X,Y):\quad\overline{A}\text{ compact}\implies\overline{\tau(A)}\text{ compact}$$ Is this true and why?
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An example of second conjugate

we know that $X$ is a normed vector space and $X'$ is the conjugate of $X$ (set of linear bounded functional defined on $X$) and $X''$ is the second conjugate of $X$ (i.e. the set of of linear bounded ...
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31 views

Operator norm of $P[v]-P[w]$

Let $\mathcal H$ be a complex Hilbert space with inner product $\langle\mid\rangle$, (dirac notation) which is semi-linear (conjugate linear) in the first argument. Let $\mathcal P_1=\mathcal P_1 ...
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1answer
48 views

Operators whose spectrum has a finite number of connected component

Assume that $H$ is a separable Hilbert space. Let $Q$ be the set of all operators$T \in B(H)$ such that the spectrum of $T$ has a finite number of connected component. Is $Q$ a subvector space ...
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Compact diagonal operator

Suppose $A : H \to H$, where $H$ is a Hilbert space, is bounded. Also, $A$ is a diagonal operator with diagonal $\{a_n\}$. Show: If $A$ is compact, then $a_n \to 0$ as $n \to \infty$. Should I prove ...
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$A^2$ has a fixed point implies $A$ has also a fixed point

Let $X$ be a Banach space and $A:X\to X$ a bounded operator. Assume that $A^2$ admits a fixed point in $X$ i.e. there exists $x_0\in X$ such that $A^2x_0=x_0$. Does this mean that $A$ has also a fixed ...
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Finite-rank operators

If $K : X \to X$ is a Banach space and $F$ is a finite rank operator (so $R(F)$ is finite-dimensional), how can I show that $KF$ and $FK$ are finite-dimensional?
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Is there an error in the solution for this exercise?

I have this exercise: H is a complex hilbert space. And T is a compact operator on H. Show that if H is not separable, then 0 is an eigenvalue of T. Hint: Use lemma 1, and theorem 2. The ...
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Finite Rank Operator: Continuity

I keep forgetting it, so... Given Banach spaces $X$ and $Y$. Then it is wrong: $$\dim\mathcal{R}F<\infty:\quad F\in\mathcal{L}(X,Y)\implies F\in\mathcal{C}(X,Y)$$ Can I construct such?
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isomorphism between function space and complex matrices

How would you show that $\mathcal{L}(X) \cong \mathbb{C}^{n \times n}$, where $X= \mathbb{C}^{n}$. Note that $\mathcal{L}(X)$ denotes the space of linear bounded functions on $X$. Is this a specific ...
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Proof of the integral operator in $L^2(\mathbb{R})$ being self-adjoint “by hand”

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(x) \, dy$$ This operator is bounded and $\|A\|=1$ (see Norm of the ...
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Normal Operators: Superalgebra (II)

Problem highlighted at the end! Application Reduction to only one operator!! Reference This builds up on: Superalgebra (I) Convention All operators possibly unbounded!! Structures Given a ...
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42 views

Find inverse operator

Let $D=\dfrac{d}{dx}.$ Consider the operator $$ D_{h,x}=\frac{e^{hD}-1}{h}. $$ Question. What is explicit form of the operator $D^{-1}_{h,x}?$ I think that $$ D^{-1}_{h,x}=\frac{h}{e^{hD}-1}, $$ ...
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Quasi ideal sequence in $B(H)$

According to comments by Hamza I revise the question. Let $H$ be an infinite dimensional separable Hilbert space. Is there an increasing sequence of subvector spaces $V_{1} \subsetneq V_{2} ...
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Why is this a bounded operator?

Let $\mathcal{H}$ be the Hilbert space $l^2(\mathbb{N})\otimes l^2(\mathbb{Z})$. I want to prove that the operator $T$ defined by $$T:=\sum_{k=1}^{\infty}{\sqrt{1-q^{2k}}e_{k-1,k}\otimes 1}$$ is a ...
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How to solve differential equations for linear operators?

I want to solve the differential equation $$ BA = \frac{\partial}{\partial t} A $$ for $A$. Here $A : H_1 \mapsto H_2$ and $B : H_2 \mapsto H_2$ are operators and $H_1, H_2$ are some Hilbert spaces. ...
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Abelian Algebras: Generator

Given a Hilbert space $\mathcal{H}$. Consider normals: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Denote their algebra: ...
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Why is $ab=ba=a^\ast b=ab^\ast=0$ (orthogonal elements in a $C^\ast$-algebra)?

Let $a,b$ be elements in a $C^\ast$-algebra $A$, such that $$a^\ast ab^\ast b=b^\ast ba^\ast a=0$$ $$a^\ast abb^\ast=bb^\ast a^\ast a=0$$ $$aa^\ast b^\ast b=b^\ast b aa^\ast =0$$ $$aa^\ast ...
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composition and strong limits of completely positive maps is completely positive

I have two claims about completely positive maps. Let $X$, $Y$, $Z$ be $C^\ast$-algebras. 1) Let $f:X\to Y$ and $g:Y\to Z$ be completely positive maps. I want to know, why $g\circ f$ is completely ...
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25 views

Dense Operators: Spectrum

This thread is Q&A. Given a Banach space $E$. Consider closed operators: $$T:\mathcal{D}(T)\subseteq E\to E:\quad T=\overline{T}$$ Then for the domain: ...
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1answer
124 views

Spectrum of weighted shift operator

The Banach space considered is the following: $(l^{\infty}(\mathbb{Z}), \|\cdot\|_{*})$ with $\|x\|_{*}=\|(...,x_{-1},x_{0},x_{1},...)\|_{*}=|x_{0}|+\text{sup}_{k\neq 0}|x_{k}|$. Define $A$, an ...
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Reference for solving linear operator equations

I'm interested in solving an equation of the form $$ Ax = b $$ for some bounded linear operator $A: H_1 \mapsto H_2$ where $H_1, H_2$ are some Hilbert spaces. I've seen in this math.SE post in ...
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41 views

Continuous functional on the linear operator

Let $\Pi, \hat \Pi$ be two linear operators from $U$ to $V$. The norm-distance is defined as $$||\hat \Pi- \Pi||=\sup_{x\in U}\frac{||(\hat \Pi- \Pi)x||}{||x||}$$ Let us define a continuous bounded ...
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35 views

Direct Sum: Stone

Problem Given Hilbert spaces $\mathcal{H}_\alpha$. Consider Hamiltonians: $$H_\alpha:\mathcal{D}H_\alpha\subseteq\mathcal{H}_\alpha\to\mathcal{H}_\alpha:\quad H_\alpha=H_\alpha^*$$ And their ...
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Is the identity in unital, simple, purely infinite $C^*$-algebra always infinite?

I'd like to prove that the identity, $I$, of a unital, simple, purely infinite $C^*$-algebra is always an infinite projection. What I'm hoping is that the following is true: If $p$ in $\mathfrak{A}$ ...
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56 views

Square root of a compact normal operator

Halmos expresses below problem in his book; Problem: If $A$ is a normal operator and if $A^n$ is compact for some positive integer $n$, then $A$ is compact. I have an example in my mind which I ...