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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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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4answers
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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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30 views

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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20 views

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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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
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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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1answer
30 views

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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1answer
40 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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32 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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1answer
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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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1answer
37 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 ...
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How to prove this “local invertibility” theorem for bounded linear operators?

The theorem states that, suppose $X,Y$ are complete normed vector spaces, if $\mathscr A_0\in \mathscr L(X;Y)$ is invertible (i.e., $\exists \mathscr A_0^{-1}\in\mathscr L(Y;X)$ s.t. $(\mathscr ...
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1answer
41 views

The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor product

Suppose we are given a C$^*$-algebra $A$ and a family of C$^*$-ideals $\mathfrak{I}$ that is upwards directed when ordered by reverse inclusion (i.e. for any $I_1,I_2\in\mathfrak{I}$ there exists a ...
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Trace: Independence

Problem Given a Hilbert space $\mathcal{H}$. Consider an operator: $$A\in\mathcal{B}(\mathcal{H}):\quad\operatorname{Tr}|A|<\infty$$ Regard ONB's: ...
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1answer
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Are they true these generalizations from matrices to operators about functional calculus?

Motivation: If we have some real function $f$ defined on an interval $I$ and $D=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$ is a diagonal matrix such that $\lambda_i \in I$ for all $1\leqslant i ...
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Physical meaning of Rudin's equation in Hilbert space

Rudin's Functional Analysis, p. 334, Corollary of Theorem 13.10 says Corollary If $a\in H$ and $b\in H$, the system of equations $$-Tx+y=a$$ $$x+T^*y=b$$ has a unique solution with $x\in ...
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Prob. 10, Sec. 4.5 in Kreyszig's Functional Analysis: How to relate this result to solution of equations?

Let $T \colon X \to Y$ be a bounded linear operator, where $X$ and $Y$ are normed spaces, both real or both complex; let $B$ be a subset of the dual space $X^\prime$ (i.e. the normed space of all the ...
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55 views

What is the precise difference between functions and operators?

I have heard affirmatively that all operators are functions, but not all functions are operators. But at the same time I have heard that functions map numbers to numbers, whereas operators map ...
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29 views

Does for $T \in B(X)$ with $\|T\|>1$ exist $T^{-1}$?

Is it true if $\|T\|>1$, where $T \in B(X)$ for some Banach space $X$, then $T^{-1}$ exists? I suppose that for $\|T\|=1$ this isn't true? Because, if we suppose that inverse exists for such ...
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Conditions for an operator on a Hilbert space to have an orthonormal set of eigenfunctions

I'm working on a problem that requires the following operator, $A^TA$, to have an orthonormal set of eigenfunctions. Note $A:H_1 \mapsto H_2$, where $H_1$ and $H_2$ are separable Hilbert spaces. ...
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1answer
40 views

Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$

I am wondering is it next true: Suppose that $f(t)$ is non-negative and non-decreasing function on $[0,\infty)$ and let $A$ be a positive operator on some infinite-dimensional separable Hilbert ...
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What is the $w^{*}$-closure of the finite rank operators in $B(H)$?

I know that the norm closure of the finite rank operators on a Hilbert space is the compact operators $K(H)$. I've been trying to determine what is the $w^{*}$-closure but I am not getting any good ...
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Ordering : Ranges

Given a Hilbert space $\mathcal{H}$. Denote selfadjoints: $$\mathcal{S}(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):A=A^*\}$$ Note that one has: $$\Delta ...
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Can I always write a bounded operator $T$ as $T=R^{*}S$

If $K$ and $H$ are Hilbert spaces and $T\in B(K)$, can I always express $T$ as a linear combination of products $R^{*}S$ for $R,S \in B(K,H)$ ? I think I already showed this is true when $K$ and $H$ ...
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How to represent non-linear operators computationally?

I have a finite dimensional vector space V, and want to compute a non-linear operator $R: V \rightarrow V$. I want to have a "general" form of this operator R. I think of the following series ...
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Uniform Boundedness: Nets

I thought so far that the uniform boundedness principle applies (according to the proof I know) to any net of bounded operators. Now I read that this works for sequences only. Can you shortly explain ...
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Existence of unique solution in Banach space

Let $X$ be a Banach space and let $L : X → X$ be a bounded linear operator. Are there situations where $||L||>1$ for which there is a unique solution to $x=Lx+b$? Explain your answer. My attempt: ...
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Norm of a product of projections in a Banach space

Let $X$ be a Banach space and let $P_1,P_2$ be two projections in $B(X)$, i.e., $P_1^2 = P_1, P_2^2=P_2$. My question: under what conditions do we have that $\Vert P_1 P_2 \Vert = \sqrt{\Vert P_2 ...
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Projections: Beppo Levi

Given a Hilbert space $\mathcal{H}$. Consider projections: $$P_\lambda\in\mathcal{B}(\mathcal{H}):\quad P_\lambda^2=P_\lambda=P_\lambda^*$$ And directed indices: ...
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Norm of the integral operator in $L^2(\mathbb{R})$.

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(y)dy$$ To find $\|A\|$ we can use the unitary Fourier transform $F$, ...
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1answer
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Example of non-normal operator whose all eigenvalues are real

Does there exist a non normal operator whose all eigenvalues are real.
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How can I solve the following exercise [closed]

Prove that a linear operator $T:X\rightarrow Y$ is bounded if and only if it maps sequences that converge to zero to bounded sequences .
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Spectral Measures: Pushforward

This thread is Q&A. Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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1answer
31 views

Continuity of multiplication of operators in the strong operator topology - find an error

I need help in finding the mistake in the following reasoning. I proved that if dimension of Banach space $X$ is infinite, then multiplication of bounded operators is separately continuous but not ...
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How exactly does one define the “spectral measure” of an operator?

I am seeing kind of different definitions of "spectral measure" at different places and its not clear to me as to what is the universal idea. It would be great to get some "standard" definition. In ...
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Stability of ground state under positive (not relatively bounded) perturbations

This is about positive perturbations that are not necessarily relatively bounded, but where the perturbed operator is known (by some independent proof) to be self-adjoint. Is this a known result (or ...
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1answer
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Normal Operators: Von-Neumann

Given a Hilbert space $\mathcal{H}$. Consider normal operators: $$N:\mathcal{D}N\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Regard their algebra: ...
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1answer
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Normal Operators: Examples

Given the Hilbert space $\mathbb{C}^2$. Consider bounded opertors: $$N:\mathbb{C}^2\to\mathbb{C}^2:\quad\|N\|<\infty$$ Then there are some with: $$N\neq N^*\quad N^*N=NN^*$$ What examples are ...
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dissipative operator meaning

Can someone explain to me the "meaning" of the dissipative operator ? https://en.wikipedia.org/wiki/Dissipative_operator I am a bit confused. Thanks in advance.
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Are lattice operations in set of orthogonal projections in Hilbert space continous?

Let $H$ be Hilbert space and denote set of all orthogonal projections in $H$ by $\Pi$. Then $\Pi$ can be given structure of a lattice. We partially order it by declaring $P \leq Q$ if $Q-P$ is ...
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1answer
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Sobolev spaces, extensions and embeddings

I have the following statement whith an argumentation which I do not understand. Fix integers $k,l$ such that $0\leq l\leq k$. Then the identity map on $C^\infty(\mathbb{T}^d)$ extends to the ...
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Closed convex hull of unitaries

If a C*-algebra ${\cal U}$ contains a non-unitary isometry $S$, show that $$\|S-A\|>\frac{1}{2n}$$ for every $A=\sum_{i=1}^n \lambda_iU_i$ which is the convex combination of $n$ unitaries. Thanks ...
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Show that a subspace is closed in a Hilbert space $H$

If $T$ is a bounded linear operator in a Hilbert space $H$, and $T$ is self-adjoint and is equal to its inverse, how can I show that $\widehat{H} = \{h + Th : h \in H\}$ is closed? If I consider the ...
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1answer
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Normal operators on a hilbert space over the reals - does $norm(Tx)=norm(T^*x)$ imply $T$ normal?

The title states the question. It's easy to prove the result for scalars C via polarisation identities but I don't think the same method works in the real case: Let $S=TT^*-T^*T$ then one obtains ...