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 $C_0-$group generated by the operator $(Af)(x)=f'(x)+a(x)f(x)$

Consider the Banach space $L^1(\mathbb{R})$ of integrable functions $f:\mathbb{R}\to \mathbb{R}$. Consider the unbounded operator $A$ defined by $$(Af)(x)=f'(x)+a(x)f(x), \ \ \ x\in \mathbb{R}$$ for ...
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34 views

Quadratic Functional Differentiability

I would like to solve the following: Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. Consider the quadratic functional $\Phi$ defined by: \begin{equation} ...
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57 views

How to calculate the adjoint of an operator and its domain?

Let $A : D(A) \subset L^2(0,1) \to L^2(0, 1)$, $$D(A) = \{u \in H^2([0, 1]) : u(0) = u'(1) = 0\}$$ $$Au = u''.$$ Can someone explain how to calculate the adjoint of A, $A^*$, and the domain of $A^*$, ...
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Showing that a domain of an operator is dense in $L^2$

Let $A : D(A) \subset L^2(\Omega) \to L^2(\Omega)$, where $$D(A) = \{u \in H^2([0,1]) : u(0) = u_x (1) = 0\}.$$ Show that $D(A)$ is dense in $L^2((0, 1))$. $D(A)$ is dense in $L^2((0, 1))$ if ...
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an operator question

I know how the derivative operator $\Big(\frac{d}{dx}\Big)^n$ works. But then how does it work if I have $$\exp{\Big(a\frac{d}{dx}+b\frac{d^2}{dx^2}\Big)}f(x)$$ I thought to use $$\exp ...
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62 views

Operator matrix is invertible if and only if its determinant is invertible

Let $A,B,C,D$ are pairwise commutative operators on a Hilbert space $H$, then a necessary and sufficient condition that the operator matrix $$\begin{pmatrix} A&B\\C&D\end{pmatrix}$$ be ...
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52 views

Does this show that it is a bounded linear operator?

Let $X$ be a Hilbert space and $A\in\mathcal{L}(X)$. I want to show that $\displaystyle e^{At}:=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}=T(t)$ defines a strongly continuous semigroup (i.e. a ...
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95 views

Differential of an operator $\phi: Mat_{2 \times 2}{\mathbb{R}} \rightarrow Mat_{2 \times 2}{\mathbb{R}}$

Let's consider an operator $ \phi: Mat_{2 \times 2}{\mathbb{R}} \rightarrow Mat_{2 \times 2}{\mathbb{R}}$ so that $A \rightarrow A^{-1}$. How to evaluate its differential? By the differential we ...
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156 views

The closed unit ball is not compact in infinite dimension spaces. Why?

We know that in finite dimension spaces the closed unit ball is compact, that is if H is a finite dimension space, then there exists an $u$ in the closed unit ball in H and $T \in \mathcal{L}(H, ...
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40 views

Calculating a norm of an operator

Let $T \in (C([a, b]))^*$, $$ T(u) = \underset{a}{\overset{(a+b)/2}\int} u(x) dx - \underset{(a+b)/2}{\overset{b}\int} u(x) dx. $$ Show that $ || T || = b - a $. We have that $$|| T || = ...
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22 views

Application of Uniform Bounded Principle (UBP)

Let $Y$ be a Banach space, and $Z$ be a n.v.s. If $(B_n)_n\in L(E,F)$ with the property that for all $(y_n)_n\in Y$, that $\|y_n\|\rightarrow 0$, we have $\|B_n(y_n)\|\rightarrow 0$. Prove that ...
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42 views

adjoint of an operator. on $L^2(0,1)$, $Bf(x)=\int_0^x f(t)dt$

I see that the above operator is bounded. I ended up with an argument to calculate the adjoint as follows, $$ <f,Bg>=\int_0^1\overline{f(x)} \int_0^xg(t)\,dt\,dx $$ I see $f(x)$ as the ...
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30 views

Extension of a self adjoint Operator

Suppose we have a open (bounded) domain $\Omega$ in $\mathbb R^d$. And let a plane $\mathcal P$ in $\mathbb R^d$ divides the domain in two (disjoint) open sets. (say $\Omega_1$ and $\Omega_2$) Hence ...
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A sequence of strongly continuous one-parameter unitary groups

Suppose that for a sequence $\{A_n\}_n$ of bounded self-adjoint operators in a Hilbert space $\mathcal H$ we have $e^{itA_n} \to e^{itA}$ strongly, for all $t \in \mathbb R$, where $A$ is a (possibly ...
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Discrete and Essential spectrum of Laplacian in $\mathbb R_{+}$ (with weird boundary conditions)

I am given on Hilbert Space $\mathcal H=L^2(\mathbb R_{+})$ $$ Af(x)=-f''(x) $$ and Domain of A is $$ D(A)=\{f\in H_2(\mathbb R_{+})\;\;| \;\;f'(0)+\alpha f(0)=0\} $$ for some $\alpha \in ...
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Relating Fourier transform theory on two distinct subspaces

In Fourier transform theory (on $\mathbb{R}$), three vector spaces play a very important role: $L^1(\Bbb R)$, $L^2(\Bbb R)$ and the Schwartz space $\mathcal{S}(\Bbb R)$. Arguably the nicer spaces of ...
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102 views

how to prove this epsilon-delta property for continuous functional calculus with normal elements?

Let $ A$ be a C* algebra, $f\in C([-1,1])$. Prove that for every $\epsilon >0, \exists \delta >0,$ s.t. for $\forall x \in A, x=x^*, \| x \| \leq 1$ and $\forall y \in A, \|y\| \leq 1$, we have ...
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76 views

Compact operators on Hilbert Space

I m working on the following problem: Let $K:H\rightarrow H$ be a compact operator on a Hilbert space. Show that if there exists a sequence $(u_n)_n\in H$ such that $K(u_n)$ is orthonormal, then ...
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61 views

Resolvent: Decay Behavior

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote resolvent set: ...
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150 views

Eigenvectors of operators on a tensor product Hilbert Space

Suppose I have finite dimensional Hilbert spaces $V$, $W$, and an operator $A$ acting on vectors in $V$ such that it has eigenvectors/values $Ax_a=\lambda_ax_a$. In the tensor product space I want to ...
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Spectral Measures: Core Lemma

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a dense domain: ...
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113 views

Spectral Measures: Scale Embeddings

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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207 views

Convergence of the spectrum under norm resolvent convergence

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in norm resolvent sense. Since $A_n \to A$ in strong ...
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49 views

Orthonormal system of simultaneous eigenvectors

Suppose we have a commutative family of compact, self-adjoint operators on a Hilbert space. Prove that there is an orthonormal system of simultaneous eigenvectors for the family. I'm not sure how to ...
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38 views

Discrete Derivative: Closure?

Problem Given the Hilbert space $\ell^2(\mathbb{N})$. Consider the operators: $$T_0:\ell^2_0(\mathbb{N})\to\ell^2(\mathbb{N}):\quad T_0(a_k)_k:=(ka_k)_k$$ ...
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35 views

Zero Tensor Product

Suppose we have a space $|\psi_1\rangle \otimes |\psi_2\rangle \otimes |\psi_3\rangle$, and operators (matrices) A ⊗ B ⊗ C acting on this Hilbert space (like in quantum mechanics). I'm trying to ...
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43 views

Spectral Measures: Analytic Elements

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote the convergence radius by: ...
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45 views

Sot closure of the unit ball of a subalgebra of $B(H)$

Let $A$ be a $C^* -$ subalgebra of B(H) and $S$ be the closed unit ball of $A$. 1- $S$ is convex and bounded, so $ S = weak^* -cl ~S$. (Is it correct?) 2- By the Kapalansky density theorem, we have ...
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WOT convergence in the unit ball of B(X)

My questions is (probably) related to: On separable Hilbert space $H$, weak operator topology is metrizable on bounded parts of $B(H)$ Does the theorem quoted in the above question, together with ...
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33 views

Spectral Measures: Nelson

Problem Given a Hilbert space $\mathcal{H}$. Consider a symmetric operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad T\subseteq\overline{T}\subseteq T^*$$ Denote the convergence radius by: ...
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Compactness of unit ball in WOT of B(X)

It is known that the unit ball in $\mathcal{B}(H)$, where $H$ is a separable Hilbert space is compact in the weak operator topology. Is it the same true if instead of $H$ we have any separable Banach ...
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53 views

Proof of von Neumann's Theorem about dense domain

The theorem due to von Neumann is the following: if $T$ is a closed densely defined Operator with domain $D(T)$, then also $D(T^*T)$ is also dense. I am searching for one proof of this non-trivial ...
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1answer
51 views

Reducing Spaces: Characterization

Given a Hilbert space $\mathcal{H}$. Consider an operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad\mathcal{D}:=\mathcal{D}(T)$$ Regard a subspace: ...
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Reducing Spaces: Domain

Problem Given a Hilbert space $\mathcal{H}$. Consider normal operators: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Denote for readability: ...
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Proving projection operators are bounded

Suppose $X$ is the direct sum of two normed linear spaces $X_1,X_2$, ($X=X_1 \dot{+} X_2$). Then I need to prove the projection operators $\Pi_1:X \to X_1$ and $\Pi_2:X \to X_2$ are bounded ...
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Singular values of compact operators

Let $T$ be a compact operator.Then show that $\sum_{n=1}^\infty$$M_n(T)=\sup\{||PT||_1:P\mbox{ is a rank } N\mbox{ projection}\}$ where $M_n$'s are the singular values of $T$ and ...
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131 views

Prove the operator is positive

I'm searching for an alternative proof of the following: Let $U$ be a self-adjoint operator on a Hilbert space $H$, define $m=\inf_{\|x\|=1}\langle Ux,x\rangle$ and $M=\sup_{\|x\|=1}\langle ...
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About closed linear operators defined on a subset of a Banach space into a Banach space?

I have this weird theorem here: If $A$ is a closed linear operator from Banach space $X$ into Banach space $Y$ i.e. $A:X\rightarrow Y$ (please take note that $A$ is closed but not necessarily ...
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Interpreting the lingo of a definition

The Terms I grew up with: A bounded linear operator $U$ on a Hilbert space $H$ is a partial isometry if there exists a subspace $M$ of $H$ such that $\|Ux\| = \|x\|$ for all $x\in M$, and $Ux = ...
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Heat operator formalism via spectral projections and Dirac measure

I am currently reading very helpful notes on the Heat kernel $p(t,x,y)$ on a Riemannian manifold $M$ -- there is one aspect though that I am not sure I understand notationally, it says that we can ...
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How to show Legendre Operator $L_{m}=-\frac{d}{dx}(1-x^{2})\frac{d}{dx}+\frac{m^{2}}{1-x^{2}}$ is Selfadjoint?

Let $m$ be a positive integer and define $$ Lf = -\frac{d}{dx}(1-x^{2})\frac{df}{dx}+\frac{m^{2}}{1-x^{2}}f $$ on the domain $\mathcal{D}(L)\subset L^{2}(-1,1)$ consisting of all twice ...
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Questions about the Kapalansky density theorem

I'm studying Takesaki's Theory of operator algebras book by myself. The following is a theorem from that book: I have several questions about this proof: 1- He claims, in the first line of ...
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A property of exponential of operators 2

Let $X$ be a Banach space. The other day I asked if all bounded operators $A:X\to X$ satisfy the following property: (P): All bounded nonzero trajectories $t\mapsto e^{tA}x$ satisfy $$\inf_{t\in ...
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Is adjoint operator of generator of an analytic semigroup be a generator of analytic semi-group?

Let $X$ be a Banach space. The adjoint semigroup $\{T(t)^\prime:t\ge 0\}$ consisting of all adjoint operators $T(t)^\prime$ on the dual space $X^\prime$ is, in general, not strongly continuous where ...
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1answer
61 views

Isolated eigenvalues of a self adjoint operator

If $X$ is a separable Hilbert Space and $T : X \to X$ selfadjoint and bounded, then the point spectrum $$ \sigma_p(T) $$ is only countable as explained here. I have the following three questions: ...
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Mourre Theory: Resolvent Formula

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote its resolvent by: $$z\in\rho(H):\quad R(z):=(z-H)^{-1}$$ Introduce its ...
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103 views

Is the Norm of the Square Root of an Operator equal to the Square root of the Norm of the Operator

Suppose we have a positive operator $A \in \mathcal{B}(\mathcal{H})$, does it follow that $$\|A\|^{1/2} = \|A^{1/2}\|?$$ If not, is there some relation between these quantities?
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Definition of operator norm

I want to show $T=d/dx$ is unbounded on $C^1[a,b]$ with $b>1$. Take a sequence $f(x)=x^n$, and $\|T\|=\sup_{x\in[a,b]}\frac{\|Tx\|}{\|x\|}=\frac{\|n\cdot b^{n-1}\|}{\|b\|}$. I want to claim as $n$ ...
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Why $\|f-g\| \leq \sup_{h\in H}\frac{\|h\|}{\|Kh\|}\|K(f-g)\|$?

Let $f,g\in L^2$ with Lebesgue measure. and $K:L^2\to L^2$ be some linear and continuous operator. Show that $$\|f-g\| \leq \sup_{h\in H}\frac{\|h\|}{\|Kh\|}\|K(f-g)\|$$ where $h\in H\subset L^2$.
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Tensor product of $C^*$- algebras

We know from the paper of Douglas and Howe (enter link description here) that the commutator ideal $\mathcal{I}$ of $\mathcal{A}(C(T^2))$, the $C^*$-algebra generated by Toeplitz operators with ...