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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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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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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2answers
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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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1answer
54 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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1answer
45 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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1answer
62 views

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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2answers
83 views

Spectral Measures: Scale Spaces (II)

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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1answer
90 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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1answer
34 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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1answer
34 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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23 views

Assuming $A$ is a nonexpansion in some norm, in what norm is $A^\top$ a nonexpansion.

Consider a matrix $A \in \mathbb{R}^{n \times n}$. Consider the vector norm $\| \cdot \|_\triangle = \| F \cdot \|_1$, where $F \in \mathbb{R}^{n \times m}$ and we have $m < n$ and $F$ has ...
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1answer
26 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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28 views

Riccati Finite Difference Equation:

Question: Consider the finite difference equation in Zeilberger notation: $$D_{1,x} [y] = a_0(x) + a_1 (x) y + a_2 (x)y^2 $$ Which in functional form is: $$ y(x+1) - y(x) = a_0(x) + a_1 (x) y + ...
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1answer
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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1answer
32 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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39 views

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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1answer
29 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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11 views

Compactness of unit ball in WOT of B(X), with $X$ reflexive [duplicate]

This is connected with the question below: 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 ...
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1answer
40 views

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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1answer
40 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
38 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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1answer
62 views

Reducing Spaces: Domain

Problem Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Denote for readability: $$\mathcal{D}:=\mathcal{D}(N)=\mathcal{D}(N^*)$$ ...
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33 views

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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1answer
70 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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16 views

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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34 views

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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1answer
34 views

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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2answers
64 views

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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Self adjoint operator with countable eigenvalues

It has been explained here that any self adjoint operator on a seperable Hilbert space inhabits at most countable many eigenvalues. Now I wonder, can one construct an operator $$ T : L^2([0,1]) \to ...
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1answer
48 views

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 {$T(t)^\prime:t≥0$} 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
27 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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1answer
34 views

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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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 ...
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3answers
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Why is $\partial_z\partial_{\bar z}=\frac14\left(\partial_r^2+\frac1r\partial_r+\frac1{r^2}\partial_{\theta}^2\right)$?

I have to show the identity I wrote in the title: it should be $\partial_z\partial_{\bar z}=\frac14\left(\partial_r^2+\frac1r\partial_r+\frac1{r^2}\partial_{\theta}^2\right)$ but some computation ...
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A question involving normed spaces and strictly convex spaces

Let $(X, \| \cdot \|_X)$ be a normed space and let $\| \cdot \|$ be a norm on $X$ such that $(X, \| \cdot \|)$ is strictly convex. How can I find a strictly convex space $(Y, \| \cdot \|_Y)$ and a ...
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1answer
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A question involving norms

Let $(X, \| \cdot \|_X), (Y, \| \cdot \|_Y)$ be normed spaces and $T : X \rightarrow Y$ a bounded operator. Let $x, y \in X$ and let the norm on $X$ $$ \|x\| = \|x\|_X + \|Tx\|_Y. $$ I can't show that ...
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1answer
26 views

If $\sup_{T\in \tau}|y^*(Tx)|<\infty$ then $\tau$ is bounded in $L(X,Y)$

Let $X$ be a Banach space, $Y$ a normed vector space and $\tau\subset L(X,Y)$. Show that if $\sup_{T\in \tau}|y^*(Tx)|<\infty$ for all $x\in X,y^*\in Y^*$ then $\tau$ is bounded in $L(X,Y)$. ...
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Norm of the product of an isometry and a bounded operator

Let $A$ be a bounded operator and $V$ a linear isometry, both defined on a complex Hilbert space $H$ (infinite dimensional). I could easily prove that $\|VA\|=\|A\|$. But, I just couldn't prove that ...
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1answer
48 views

Order zero maps in matrix algebra

Let $a$ and $b$ are two elements in a $C^*$algebra $A$. We say $a\perp b$ if $ab=ba=a^*b=ab^*=0$. We say a completely positive map $\phi: A \rightarrow B$ is of order zero if for any positive elements ...
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42 views

Justifying an equality involving a closed operator $A$

Justify the equality $$A \int_0^\infty e^{-\lambda t} S(t) u \, dt = \int_0^\infty e^{-\lambda t} AS(t) u \, dt$$ used in (16) of §7.4.1. (Hint: Approximate the integral by a Riemann sum and recall ...
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1answer
36 views

computation with polar decomposition of bounded operator on hilbert space

I am trying to prove the following homework problem: Let $T \in B(H)$ (so $T$ is a bounded operator on a Hilbert space $H$), and let $T = U|T|$ be the polar decomposition of $T$. Prove that if $T$ is ...
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Properties of functional calculus

Suppose we have a self-adjoint bounded operator $S$ on a Hilbert space $\mathscr{H}$ with the property that $||Sx||<||x||$ for each $x\in\mathscr{H}\setminus\{0\}$. Now assume that ...
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1answer
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Trying to show that $(c_0, \| \cdot \|_s)$ is strictly convex, where $\| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |$

I'm trying to show that $ (c_0, \| \cdot \|_s) $ is a strictly convex space, where $$ \| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |,$$ $ x = (x_1, x_2, ..., x_i, ...) \in ...
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2answers
54 views

Properties of resolvent operators

I am asked to prove the identities of $(12)$ and $(13)$, which are given on page 438 of the textbook PDE Evans, 2nd edition as follows: THEOREM 3 (Properties of resolvent operators). (i) If ...