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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Bounded Operators, Unitary group

It's clear to me that if H is a self-adjoint bounded operator on a Hilbert space, then the bounded operators $$U_t :=\sum_{n=0}^\infty (iHt)^n / n!$$ are unitary for all $t\in \mathbb{R}$. How do I ...
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Uniqueness of element in infinite dimensional Hilbert space

Suppose $H$ is an infinite Hilbert space where $\{e_k:k\in \mathbb{Z}\}$ is a total orthonormal family. Let $H_1=\overline{span{(e_k: k=0, 1,2,\cdots})}$ and $H_2=\overline{span{(e_{-k}+ke_k: ...
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What are the projections of a commutative C* algebra?

I am aware that the commutative C* algebra is $C_0(X)$ for some nice space $X$ but I cannot figure out what the projections should be. The natural candidates (indicator functions on nice subsets of ...
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Under what conditions are such operators well defined?

Let H be a hilbert space, and $\phi_k$ a basis, one can define a "diagonal" operator $A$ by $A\phi_k=b_k\phi_k$, Is there a simple condition on the coefficients $b_k$ such that the operator is well ...
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Index of an element in C*-algebra

Suppose that $x$ is an element of abstract $C^*$-algebra $A$. For example if $x$ is normal, i.e. $x^*x=xx^*$ then if we use any representation $\pi$ of $A$ on some Hilbert space $H$ then $\pi(x)$ will ...
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$P+Q-PQ$ is a projection if and only if $PQ=QP$.

Let $\mathcal H$ is a Hilbert space and $P,Q:\mathcal H \to \mathcal H$ are projections. I want to show that $P+Q-PQ$ is a projection if and only if $PQ=QP$. If $PQ=QP$ clearly $P+Q-PQ$ is a ...
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In what sense are compact operators limits of finite-rank operators?

The convergence is in respect to what topology ? Can someone please write it mathematically ?
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How to calculate the norm of this operator?

Let $H$ be a separable Hilbert space and $(\phi_k)$ be a basis $A(t)$ is defined such as $A\phi_k=\exp(-t/k)\phi_k$. I am specifically interrested whether $\|A(t)\| \to 0$ when $t \to \infty$ or not, ...
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Equivalence and rank equivalence

Let $A$ be a $*$-algebra. Let $P(A)=\bigcup_{n=1}^\infty\{p\in M_n(A):\text {$p$ is a projection} \}$. By projection I mean $p=p^*=p^2$. Define the an equivalence relation on $P(A)$ by $p \sim q ...
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$T$ is self-adjoint $\Rightarrow \exists$ positive $A,B$ such that $T=A-B$ and $AB=0$

I have a trouble by the following problem and I dont have any idea to solve it. can anybody give me a hint? Thanx in advance. Let $\mathcal H$ be a Hilbert space and $T:\mathcal H \to \mathcal ...
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Compact operators are orthogonally equivalent to a diagonal matrix?

On Brezis's Functional Analysis, the last question of Problem 44 (near the end of the book) reads (modified to include context) Assume that the Hilbert space $H$ is separable and $T\in\mathcal ...
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Hadamard product involving operators

If we have two matrices $A=(a_{i,j})_{i,j}$, $B=(b_{i,j})_{i,j}$ representing linear and continuous operators from $\ell^2$ to $\ell^2$, it is known that the Hadamard product of them, $A\ast ...
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Spectral Measures: Poisson

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}H\to\mathcal{H}:\quad H=H^*$$ And its spectral measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad ...
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What are useful mappings (operators) in image reconstruction

I'd like to ask the technician mates to provide some information regarding mappings and image reconstruction operators. Please, if possible, provide some articles and helpful discussions about useful ...
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Can we characterize the probability generating function as a linear operator?

For a nonnegative integer-valued random variable $X$ with $\mathbb P(X=j)=p_j$, we define the probability generating function of (the distribution of) $X$ by $$P_X(s):=\mathbb E\left[s^X\right] = ...
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Does there exist a countable basis for the space of all continuous meta operators?

Consider operators $O: \left( f: \mathbb{R} \rightarrow \mathbb{R} \right) \rightarrow (f: \mathbb{R} \rightarrow \mathbb{R})$ We define a continuous operator $O$ over an interval $[a,b]$ if: $$ ...
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Integration of complex exponential function over $\mathbb C$

Find the limit $$\lim_{z \to \infty}\int_{\mathbb C}|w|e^{-|z-w|^2}dA(w) $$ where A is area measure such that dA=rdrd$\theta$ Please help me, I did four page computation by changing to polar ...
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Integration of exponential function.

It is part of my research work in operator theory. I came across such a integration which became nightmare for me. I tried as all the possible ways. If anyone could help I would really appreciate. ...
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Bound self-adjoint operator

Assume we have a positive (so that we can take the square-root by functional calculus) self-adjoint operator $H: D(H) \subset \mathcal{H} \rightarrow \mathcal{H},$ then we can define ...
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Is there an example of a non von Neumann algebra with this property?

What is an example of a $C^{*}$ subalgebra $A$ of $B(H)$ such that $A$ contains the identity $I_{H}$ and satisfies the following properties: 1) For every $T\in A$, The orthogonal projection ...
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Do powers of contraction on Hilbert space converging to zero imply convergence of its adjoint to zero also?

In my functional analysis class I was met with the following problem: We suppose that $ \mathbb{H} $ is a Hilbert space and that T is a contraction operator on H (meaning $ ||T|| \leq 1 $ in the ...
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restrictions of closed linear operator to range of its powers

I am trying to prove that if $T$ is a closed densely defined operator on a Hilbert space(or Banach space), $\lambda$ is non-zero and $T_n$ is the restriction of T to range $R(T^n)$ for some n, then: ...
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Precedence of cross product and dot product

Which operator precedence is higher? The one of the cross product or the one of the dot product? Consider the following term: $$\overrightarrow {A}, \overrightarrow {B}, \overrightarrow {C}, ...
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A conjecture about traces of projections

Let $M_n$ denote the space of all $n\times n$ complex matrices. Define $\tau:M_n\rightarrow \mathbb{C}$ by $$\tau(X)=\frac{1}{n}\sum_{i=1}^n x_{ii},$$ where of course $X=[x_{ij}]\in M_n$. Recall that ...
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Closed operators

I was wondering whether the following statement is true or not? If $A$ is closed, then it follows from the closed-graph theorem that it is bounded iff $D(A)$ is closed. I found this in a chapter of ...
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$A$ and $A^*$ dissipative implies $D(A) \subset H$ is compact embedding

For selfstudy purpose I want to show the following: $H$ Hilbertspace, $D(A)$ dense subspace of $H$, $A\colon H \supset D(A) \to H$ linear closed dense defined operator. If $A$ and $A^*$ are both ...
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how is a compact embedding of infinite dimensional Banach spaces possible?

I'm looking at a dense defined closed operator $A\colon H \supset D(A) \to H$ with a Hilbertspace $H$ and $D(A)$ a dense subspace of $H$. In my notes there are some phrase like "if the embedding $D(A) ...
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An equivalent definition of self-adjoint operator?

Suppose the linear operator: $$\begin{array}{rcll} L:&C^2[a,b]&\longrightarrow& C[a,b]\\ &u&\longmapsto&Lu=p_0\ddot u+p_1\dot u+p_2u \end{array}$$ with $p_0,p_1,p_2\in C[a,b]$ ...
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$T+i\operatorname{Id}$ is an isomorphism for self-adjoint $T$

Let $T:H\to H$ be a self-adjoint continuous operator on a complex Hilbert space. Claim: $T+i\operatorname{Id}$ is an isomorphism and $\|(T+i\operatorname{Id})^{-1}\|\leq 1$. A few observations: ...
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A naïvely constructed extrapolation of a self-adjoint operator. Is it self-adjoint?

Let $\mathcal{H}$ be a real Hilbert space and let $A\colon D(A)\subset \mathcal{H}\to \mathcal{H}$ be an unbounded operator. Consider also a Hilbert triple $$ \mathcal{H}_+\subset \mathcal{H}\subset ...
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How to prove this integral operator is bounded

Consider the integral operator $f\to g$: $$g(s)=\int_0^\infty\frac{f(t)}{t+s}\, dt$$ The above operator is the result of applying the Laplace transform twice. 1) What is the name of this operator? ...
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Operators of order $k$ between Sobolev spaces

It may sound like tautology, but I have a problem in proving that differential operator of order $k$ is an operator of order $k$. What exactly I'm asking for? Let $M$ be a manifold and $E,F$ be two ...
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Riesz Theorem on C[K], K compact

I'm studying Riez Theorem on Kreyszig's book: "Introductory functional analysis" , it states that "Let $l$ a bounded and linear functional on $C[a,b]$ (continuous functions on [a,b]) , then $l$ can ...
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A question regarding Eigenvalues

Note: $\psi,\psi^{\dagger} :\Bbb{R} \to \Bbb{C}$ and $x, \lambda_i , \hbar, m \in \Bbb{R}$ Say we know that $\lambda_1$ is a solution to the eigenvalue equation: $$\hat{\Pi}\psi(x)= \lambda_1 \psi(x) ...
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How to check if an operator is invertible?

Let $T_1 : C[a,b] \to C[a,b]$ be an operator defined by $$T_ v(x)=\int^b _a (x-t)v(t) dt$$ where $a \leq x \leq b$ and $v \in C[a.b]$ How can you check if the operator $T_1$ is invertible or not?
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calculating the abstract index of $C(T)$

Consider the following definition in operator theory: I'm reading an example of the abstract index of $\mathcal{A}$ in Zhu's An Introduction to Operator Albebras: Here $G_0(\mathcal{A})$ is the ...
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self-adjoint and orthonormal basis

Suppose $F=\mathbb{R}$. Let $A: V\to V$ (where $V$ is a finite dimensional inner product space over $F$) so that $A=A^*$ ("self-adjoint"), then there exists an orthonormal basis of eigenvectors and ...
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trace class and nuclear operators

Maurin (http://www.mscand.dk/article/viewFile/10641/8662) defines nuclear operators like this: A linear operator $A:\mathcal{H}_1\rightarrow \mathcal{H}_2$ where $\mathcal{H}_1$ and $\mathcal{H}_2$ ...
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spectrum of a positif operator !! [closed]

i have this question : we say that an $H$ operator is positive if we have : $<u,Hu>\geq0$ $\forall u\in D(H) $ so how to prove that for a self-adjoint operator $H$ we have : $H$ positive ...
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$S=\frac{-d^2}{dx^2}$ self-adjoint operator or not?

I have this simple question : In $L^2(]0,1[)$ let $S$ be the operator defined by : $D(S)=C_c^2(]0,1[)$ and $S=\frac{-d^2}{dx^2}$ is this operator self adjoint, and how to prove it ? ($C_c$ : ...
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Finding the norm of an operator

Consider the linear operator $T : C[0,1] \to \mathbb{R}$ defined by \begin{align*} T(x) := x(0) - \int_{0}^{1} x(t)\phantom{.}dt \end{align*} Show that $T$ is bounded and find its norm ...
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Using the definition of the operator norm

I am given the following problem: Using the definition $$\lVert L \rVert_{\text{op}}=\sup_{\vec{u} \in \mathbb{R}^d, \lVert \vec{u} \rVert=1}\lVert L\vec{u} \rVert$$ of the operator norm of a ...
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Commutating nilpotent operators

Is there good examples of collection of nilpotent operators that commute with themselves? Is there a good reference for a collection commutative nilpotent operators that commute with themselves or ...
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A counter example for adjoint of unbounded operators

I need a counter example for $(A+B)^*=A^*+B^*$, where $A$ and $B$ are unbounded operators on Hilbert space and $^*$ denotes the adjoint.
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Gauge invariance of a magnetic Schrödinger operator

Good morning, I am studying the properties of the magnetic Schrödinger operator $$ \mathcal{L}_A = \left( -\mathrm{i} \nabla -A \right)^2 = \left( -\mathrm{i} \nabla -A \right)^\dagger \left( ...
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Spectral density for the operator $A u = - u''$?

How to prove that the spectral density for the operator $A u = - u''$ on the whole real line is $$ e(x,y;\lambda) = \frac{\chi(\lambda) \, cos\lambda^{1/2} (x-y)}{2\pi \lambda^{1/2}} $$ where $\chi $ ...
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Sturm-Liouville operator with Dirichlet BC

I am trying to understand why Sturm-Liouville operator $$L(f)(x)=f''(x)-p(x)f(x)$$ with Dirichlet boundary conditions on $[a,b]$ is unbounded. $f$ is twice continuously differentiable, $p(x)>0$ is ...
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Boundedness and norm of a sequence operator

Let $s = \{s_{n}\}_{n=1}^{\infty}$ be a fixed and bounded sequence of real numbers, i.e. $s \in (\ell^{\infty},\|\cdot\|_{\infty})$. Consider the operator $T_{s} : \ell^{2} \to \ell^{2}$ defined ...
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Boundedness and norm of a linear operator

Consider the linear operator $T : C[-\pi,\pi] \to \mathbb{R}$ defined by $$ Tf := \int_{-\pi}^{\pi} f(t)\sin(t)\phantom{.}dt $$ Show that $T$ is bounded and find its norm $\|T\|$. Consider ...
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Generalization of matrix inversion lemma

I am looking for an operator version of matrix inversion lemma. To be specific, does the identity also hold for operators defined on general (infinitely dimensional) Hilbert space, possibly with ...