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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Properness of isometric actions of discrete groups on affine Hilbert spaces

I've been reading Valette's introduction to the Baum-Connes conjecture and as I read the example of a construction of a (model of the) classifying space for proper actions of $\Gamma$ (discrete) given ...
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19 views

Interchanging Limit and Integral sign

I'm reading a book on composition operators, and the author makes the following claim: Given a self-map of the unit disc, and a $H^2$ function $f$, where $H^2$ is the Hardy space, if we fix a radius ...
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31 views

Does a contraction converging in power series necessarily lead to the operator being a proper contraction?

I was recently met with this in my functional analysis class on which I am stuck: Let $ \mathbb{H} $ be a Hilbert space and let T be a contraction operator on $ \mathbb{H} $ (meaning $ ||T|| ...
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33 views

Two sequences of operators on Hilbert Space

Let $H$ is some Hilbert Space, and $a_n,b_n \in B(H)$ is sequences of some operators on it. We know, that $a_n b_n$ converges to $v$ by norm. We also know, that all $b_n a_n$ are strictly positive. ...
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6 views

$L_p$ version of Toeplitz extension

There is the well-known extension of $C^*$-algebras $0\rightarrow K\rightarrow\mathcal{T}\rightarrow C(S^1)\rightarrow 0$ where $\mathcal{T}$ is the Toeplitz algebra (generated by the unilateral shift ...
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19 views

Does strict contraction lead to convergence to zero in norm?

In my functional analysis class I was asked this question which got me stuck: Let $ \mathbb{H} $ be a Hilbert space and we are given an operator T satisfying: $ || T || < 1 $ in the ...
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29 views

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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16 views

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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24 views

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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12 views

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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22 views

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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29 views

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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40 views

Spectrum of the derivative operator: What's wrong in my argument?

Consider the Banach space $X=C[0,1]$ of continuous functions $f:[0,1]\to\mathbb{R}$ equipped with the supremum norm. If we consider the following unbounded operator $A$ defined on its domain ...
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34 views

Why is $(\sqrt{P})^2=P$ where $P$ is a positive operator on a Hilbert space?

The following is a proposition regarding positive operators on a Hilbert space in Douglas's Banach Algebra Techniques in Operator Theory: Corollary 4.32 is as the following: I understand that the ...
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13 views

$C^\ast$ condition implies $B^\ast$ condition

By $C^\ast$ condition I understand $\|A^\ast A\|=\|A^\ast\|\|A\|$ and for $B^\ast$, $\|A^\ast A\|=\|A\|^2$. I know these conditions are equivalent even NOT assuming the involution is isometric, but I ...
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40 views

Do compact convergence topology and w*-topology coincide on the Pontryagin dual group of a LCA group.

Given a locally compact group $G$, consider its Pontryagin dual $G^{'}$. Do compact convergence topology and weak*-topology on $G^{'}$ coincide?. When we talk about weak*-topology on $G^{'}$, it ...
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31 views

Unbounded linear operator with bounded restriction

Given that a linear operator $T:X\rightarrow Y$, where $X$ and $Y$ are both Banach spaces, $D$ a dense subspace of $X$, if we know that the restriction of $T$ to $D$, say, $S=T|_{D}$ is bounded, then ...
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24 views

Does spectral decomposition exist for non-self-adjoint operators?

In theory, if a linear operator $P$ in a Hilbert space $H$ is self-adjoint, we can decompose it as $Pu=\sum_i \lambda_i <\phi_i,u>\phi_i$, where $\phi_i$ is the eigenfunction of $P$. And we can ...
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13 views

closability of $n$-th power of paranormal operator

It it well-known, that there exists closable paranormal operator $A$ such that $\overline{A}$ is not paranormal [1] and if $B$ is paranormal then $B^n$ is also paranormal [2]. Is there any example of ...
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24 views

concomitant and self-adjoint operator

If $Lu = u^{\prime\prime}+\omega^2u$, show that $L$ is formally self-adjoint and the concomitant is $J(u,v)=vu^\prime-uv^\prime$. Moreover, if $u$ is a solution of $Lu=0$ and $v$ is a solution of ...
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59 views

Projection operator in Banach space is continuous

Let $(X,||\cdot ||)$ be a Banach space with a Schauder basis, i.e. there exist $e_j \in X$, $j\in \mathbb{N}$, s.t. $||e_j||=1$ for all $j$ and every $x\in X$ can be uniquely represented as ...
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42 views

Proof compactness of adjoint operators

I am trying to understand a proof of the following statement: Given a complex B-space X and a compact operator $T:X\rightarrow X$, the adjoint operator $T^\ast:X^\ast \rightarrow X^\ast$ is compact as ...
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24 views

How do we compute the Green's function of the Laplacian?

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for ...
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48 views

Show that $S_n \to S $(weakly) and $T_n \to T$ strongly implies $S_nT_n \to ST$ weakly

Let $X,Y,Z$ be Banach Spaces. Let $T_n,T \subset BL(X,Y), S_n,S \in BL(Y,Z)$. Show that a) $S_n \to S $(weakly) and $T_n \to T$ (strongly) implies $S_nT_n \to ST$ (weakly) b) $S_n \to S ...
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65 views

The reduction of nilpotency order of nilpotent elements of $C^{*}$ algebras

Assume that $A$ is a unital $C^{*}$-algebra. Let $a\in A$ be a nilpotent element with $$a^{k}=0,\;\;k>1.$$ Are there two elements $x,y\in A$ with $a=xy,\;\;(yx)^{k-1}=0$? Motivation for ...
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22 views

Solvable Schrödinger operator

I am currently looking for an example of a 1-dimensional Schrödinger Operator $-\frac{d^2}{dx^2} + V(x)$ with the following properties: 1) V(x) must be integrable 2) $-\frac{d^2}{dx^2} +V(x)$ must ...
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37 views

What are eigenvalues/eigenfunctions of a “pointwise product” operator

Let us consider the Hilbert space $l^2([0,1])$ with inner product $<u,v>=\int_0^1 u(x)v(x)\mathrm dx$. We define a pointwise product operator $A$ as $(A\circ u)(x)=a(x)\cdot u(x)$, where ...
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25 views

Is there a sample-path continuous stochastic process whose sample paths do not almost surely lie in an RKHS?

Let $f$ be a mean zero second-order stochastic process with continuous covariance function $k$, that is indexed on a separable metric space $\mathcal{X}$ and that is sample-path continuous. Can we ...
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12 views

$V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 : the semi group $V(A)$ of equivalent projections (under Murray Von Neumann equivalence) in $M_∞(A)$ is ...
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44 views

Difference between Schmidt decomposition and singular value decomposition

Schmidt decomposition of an operator is a useful tool of quantum information theory nowadays. Let $O$ be an operator acting on the Hilbert space $\mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}$. ...
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31 views

Convolution operator is normal

Consider the convolution operator $$Tf(s)=\frac{1}{2\pi}\int_0^{2\pi}f(t)h(s-t)\,\,dt,\quad f\in L^2[0,2\pi]$$ where $h:\Bbb R\to \Bbb C$ is a $2\pi$-periodic function, square integrable on ...
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21 views

How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$?

Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha ...
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46 views

Eigenfunctions of integral operator

I am faced with the problem of calculating the eigenfunctions for an operator of the form: $(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy $ Does anyone know for which functions (or types of ...
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11 views

Closable Multiplication Operator

I have the operator $M:Dom (M)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$, $Mf(x)=m(x)f(x)$, where $m$ is a continuous function and $Dom(M)=\{f\in L^2(\mathbb{R}^N)| mf\in ...
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39 views

C*-algebras: Proofs on $C_0(X)$

I'm looking to prove the following but am stuck, please can you help me? $C_0(X)$ is isomorphic as a C*-algebra to $C_0(Y)$ if and only if X is homeomorphic to Y, where X and Y are locally compact ...
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7 views

Calculate the matrix of a linear opertor that transforms a vector to a Hankel matrix

I would like to calculate the matrix associated to a linear operator $\mathbf{R}$ that transforms a vector $\mathbf{x}\in\mathbb{R}^N$ into a Hankel matrix $\mathbf{H}\in\mathbb{R}^{N-Q+1\times Q}$ ...
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42 views

Extend and restriction of operator on $B(H)$

Let ‎$‎‎H$ ‎be a ‎Hilbert ‎space ,‎‎‎‎‎‎$‎‎B(H)$ ‎be ‎bounded ‎operators ‎on ‎‎$‎‎H$ ‎and ‎‎$‎‎K(H)$ ‎be ‎compact ‎operators ‎on ‎‎$‎‎H$‎. Assume ‎that ‎‎$‎‎M$ ‎is a ‎close‎d subspace of ‎$‎‎H$ ‎and ...
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25 views

Some remarks about linear operators from PMA Rudin

Let $L(X,Y)$ be the set of all linear transformations of the vector space $X$ into the vector space $Y$. For $A\in L(\mathbb{R}^n,\mathbb{R}^m)$, define the norm $\lVert A\rVert=\sup\limits_{x: ...
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an isomorphism from $L^\infty(\mathbb{T})$ to $L^\infty([-1,1],\displaystyle\frac{2}{\pi } \sqrt{1-t^2}\mathrm{d}t)$

$\mathbb{T}$ is the boundary of unit ball.Consier $\phi:[-1,1]\rightarrow\mathbb{T},\phi(t)=exp^{2i(arcsint+t\sqrt{1-t^2})},t\in[-1,1]$. It is easy to check that $L^2(\mathbb{T})\ni f\mapsto f\phi\in ...
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36 views

Operator and interpolation

I'm going to speak very informally, because i don't know how to formalize this stuff. If we have knots $X = \left\{ x_0, \ldots x_n \right\}$ and a given function $f$ belonging to an appropriate ...
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37 views

Show that an operator is negative

I would show that, the operator $$A = \left(x_{4} \frac{\partial}{\partial x_{1} } -x_{1} \frac{\partial}{\partial x_{2} } \right) \frac{\partial}{\partial x_{3} } $$ is a negative operator on ...
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59 views

Weak convergence = norm convergence for trace class operators?

Given a (separable) Hilbertspace $H$, I look at the traceclass operators $\mathfrak{S}_1$. I recall the fact that the weak convergence implies norm convergence in the sequence space $\mathcal{l}^1$. ...
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23 views

An operator which moves on the boundary

Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis in $H$. Let $E_0$ be a countable subset of $E$ and $p$ be the projection onto the space generated by $E_0$. Let ...
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29 views

Derivation of an integral equation

I have the following system $$\frac{d}{dx}\left(a(x)\frac{du}{dx}\right)=f, \text{ for } x \in (0,1)$$ with boundary conditions $u_x(0)=0$ and $u(1)=0$. For $a(x)>0$, and $b(x)=\frac{1}{a(x)}$, I ...
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35 views

Is there a mistake in the solution given to this exercise?(compact linear operators)

The exewrcise is: Let $\mathcal{H}$ be an infinite-dimensional Hilbert space, with an orthonormal basis $\{e_n\}$ and let $T \in \mathcal{B}(\mathcal{H})$. Show that if T is compact then ...
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22 views

Orthogonal, Normal, and Self-Adjoint operators

Let be $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ the operator given by $T=3ref_{x_2=2x1}$, where $ref_{x_2=2x1}$ is the reflection on the line $x_2=2x_1$. Define the operator $U:{(\mathbb{R}^2)}^n ...
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29 views

Spectral properties of operator over $L^2(0,1)$

Let be $I=[0,1]$ the unit interval and define the operator $$(Af)(x)=\int_0^x f(t)dt $$ with domain $C_0^{\infty}(I)\subseteq L^2(I)$. I want to show the following: $A$ is bounded and compact; ...
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29 views

Can we use a series of properties to determine integral operator $f \to \int_0^1 f d\mu $

Question: Suppose there exists an operator $I: C^{\infty}(0,1) \to \mathbb R$ satisfying the following properties: (1) $I (\chi_{(0,1)})=1$ ; (2) $I(kf)=kI(f)$, where $k\in \mathbb R$ and $f\in ...
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26 views

Riemann Lebesgue Lemma for locally compact ableian groups

I'm looking for a reference (or proof here) of the generalized RL Lemma for LCAGs. One result is that if $G$ is a LCAG then $$\{ \hat{f} : f \in L^1(G)\} \subset C_0(\Gamma)$$ where $\Gamma$ is the ...
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61 views

Convergence of operator-exponential

Let $T: [0,\infty) \rightarrow L(X)$ define a $C_0$ semigroup on a Banach space $X$, then I want to show that $A_h:=\frac{T(h)-id}{h}$ are such that $e^{tA_h}(f) \rightarrow T(t)(f)$ pointwise. ...