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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140 views

Half Laplace operator

I'm curious whether a half Laplacian (or square root of Laplacian) exists. More specifically, I'm looking for an $X:C^2(\Bbb R^n)\to C^2(\Bbb R^n)$ operator such that $$\forall f:XXf=\Delta f$$ I know ...
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62 views

Three basic questions about linear operator in a Hilbert space

Just come across three questions in reading a paper. Suppose we are dealing with a Hilbert space of $L_{2}[0,1]$ and all the functions mentioned below are in $L_{2}[0,1]$. Define the operator $A$ by ...
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53 views

$\gamma-$radonifying operators.

I am reading about $\gamma$-radonifying operators, and came across 2 similar definitions. And I hoped to understand why they are equivalent. Let $H$ be a seperable real Hilbertspace, $E$ banach ...
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37 views

operator differential equation

let be the differential equation for the operator 'X' $$ \frac{dX(t)}{dt} = A(t) X(t) $$ the formal solution is the exponential operator $$ X(t)=X(0)e^{ \int_{0}^{t}A(u)} $$ of course i should ...
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38 views

Measurability of the dilatation operator

I need some help with this question: We consider the dilatation operator: $T: \mathbb{R^{+}}\to \mathcal{L}(L^p(\mathbb{R}),L^p(\mathbb{R}))$ $\;\;\;\;\;\;\delta\to ...
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90 views

Self-Adjoint Operators

If an Operator $L$ is defined as $Lu=u''$ and $a_1u(0)+b_1u'(0)+c_1u(1)+d_1u'(1)=0$ along with $a_2u(0)+b_2u'(0)+c_2u(1)+d_2u'(1)=0$, then for what values of $a_1,b_1,c_1,etc$ is the operator ...
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61 views

Smoothing operator on manifolds

I am reading John Roe "Elliptic operators, topology and asymptotic methods". On page 79 there is the definition 5.20 of smoothing operators. The definition is the following: "A bounded operator on ...
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63 views

Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
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120 views

Spectral theorem and projection

This should be simple, but I'm stuck. Let $A$ be an unbounded self-adjoint operator on a Hilbert space $H$. The spectral theorem says that there is a decomposition of $H$ into a direct sum for which ...
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28 views

Piecewise Continuous Fredholm Kernel

Suppose I have a ``Fredholm equation of the second kind" with kernel $$ K(x, y) = 1[x \geq u]k(x, y) $$ where k(x, y) is continuous. Let $f_n(x)$ be a smooth continuous approximation of $1[x \geq ...
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33 views

Generator of a transport semigroup on the torus

Consider on the Banach space $C( \mathbb T;\mathbb R)$ of continuous functions on the 1-dimensional torus $\mathbb T:=\mathbb R/\mathbb Z$ (equipped with uniform norm) the operator $L$ given by $$ L ...
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37 views

Composition of analytic functions is analytic in a general setting, and are they continuous?

Regarding the notion of analyticity discussed in this setting: A possible equivalence for holomorphicity I wonder if this is truly the correct definition (even though it is from Dunford-Schwarz) An ...
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54 views

Is this operator closed?

Consider the linear operator $H$ with domain $D(H) = S(\mathbb R)\subset L^2(\mathbb R)$, where $S(\mathbb R)$ is Schwartz space, defined by \begin{align} H\psi(x) = -ix^3\frac{d\psi}{dx}(x) -i ...
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66 views

Self-adjointness of differentiation operator

Let's say $\mathcal{H}=L^2([0,1])$ and $p$ is the operator $-i\frac{d}{dx}$ defined on $\mathcal{D}(p)=\{f\in L^2([0,1])\ |\ f'\in L^2, f(1)=e^{i\theta}f(0)\}$. I have to prove that $p$ is ...
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150 views

Comparison of Strong OPerator and Weak * Topologies on B(H)

It is known that in $\mathfrak{B}(\mathbb{H})$, the weak operator topology (WOT) is contained in both the strong operator topology (SOT) and $\sigma$-weak topology. In general the SOT and the ...
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100 views

Compactness of advection diffusion operator (elliptic operator)

Consider the ADE defined on a compact domain $\Omega \in \mathbb R^n$, with boundary $\partial \Omega$ Assume $u(x)$ is smooth incompressible vector field, with $u(x).\hat n(x)=0$ for $x\in \partial ...
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86 views

Show that $(D_t t)$ is an isomorphism.

Let $B$ be a Banach space, $\epsilon > 0$, and $$C_1^0 ([-\epsilon,\epsilon],B) = \{u \in C^0([-\epsilon,\epsilon],B) : tu \in C^1([-\epsilon,\epsilon],B)\}.$$ Denote $\partial/\partial t$ by ...
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45 views

When does Gelfand Naimark Theorem Hold?

I was going through the proof of the Gelfand Naimark Theorem for the Unital Commutative Banach Algebras. In proving that each character has norm equal to $1$, we used the fact that $||e|| = 1$ where ...
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83 views

Transpose of the Hilbert-Schmidt operator

Let $X = L^2(\Omega)$, $\Omega \subset \mathbb{R}^N$ be an open set (or a $\sigma$-finite measure space), $B \in L^2( \Omega \times \Omega)$. Then the Hilbert-Schmidt operator $T \in \mathcal L(X)$ ...
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63 views

Kernel of an infinite dimensional operator

Can the Kernel of an infinite dimensional operator have $\dim=0$? I am thinking to the annihilation ($\hat E$) and creation ($\hat E^\dagger$) operators. Suppose, in fact, we have an infinite but ...
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58 views

Almost everywhere analytic function

Suppose we have a measure space $\Omega$ and a function $m\in L^\infty(\Omega,\mathcal{B}(E))$, that is invertible for almost all $\theta\in\Omega$ Further assume, that we have an other function $G$ ...
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88 views

exponential of an operator, all to a power

I saw this almost answered here: Exponential of the differential operator (it is the unaccepted answer) What I am looking to "solve" is $$ \sum_{j=0}^d\; \left( e^{\epsilon\,\partial_x} \right)^j ...
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100 views

Stampacchia Problem

I need to solve this problem, but don't know how get that particular bound. Please, somebody can help me? Let $V$ a Hilbert space, $a : V\times V\rightarrow\mathbb{R}$ a bounded bilinear form, ...
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29 views

an upper bound on Wave Front

Can you please help to understand how to solve this question: Let $f^{ij}(x)$ be a positive definite matrix smoothly varying with $x$ and define ...
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17 views

similarity between bundle shift

Let $E$ be a flat unitary bundle of rank $n$ over a domain $R$ in $\mathbb{C}$. It is known that bundle shift $T_{E}$ is similar to $T_{\mathbb{C^n}}$ (which is the bundle shift corresponding to the ...
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51 views

Relationship of two generalizations of the real/complex calculus

On the one hand, one has the various functional calculi from Operator Algebras. The continuous functional calculus for C* algebras, the bounded borel functional calculus for Von Neumann Algebras, the ...
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56 views

Prove operator $T$ is onto

Consider the Hilbert spaces $X := H^{1}(\Omega)\times H^{1}(\Omega)$ and $Y:=L^2(\Omega)\times L^2(\Omega)$, where $\Omega =\ ]{-}\pi, \pi[$, and \begin{eqnarray*} \langle(u,v), (z,w)\rangle_X & = ...
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162 views

Compact integral operator

I have this exercise and I don't know how to solve the last question. In the following $a,b$ are two real numbers such that $a<b$ ,$E=C([a,b],\mathbb{R})$ with the norm $||.||_0$ given by ...
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86 views

Find spectrum of the operator and an explicit form for the solution

Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if $v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$ (a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum (b) ...
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Find spectrum of the operator

I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem. Let $\theta \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator $Au=v,$ ...
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124 views

How can projection operators be limits of powers of unitary operators?

Consider a (fixed) unitary operator $U$ acting on the Hilbert space $\mathcal{H}$. Because the unit ball is compact in the weak topology, it is not hard to see that there exists a (smallest) compact ...
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33 views

The Square of the Laplace Transform

I have been looking at the Laplace transform $$\mathcal{L}f(s)=\int_0^{\infty}f(t)e^{-st}dt$$ and I'm trying to find The norm of $\mathcal{L}^2$ The nullspace of The norm of $\mathcal{L}^2$ So ...
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227 views

Proving properties of exponential map on a Banach algebra

$$\exp(a) := \sum\frac {a^k}{k!}$$ Can you help me prove that: $\exp$ is well defined (i.e. converges for all $a$ in $A$) $\exp$ is continuous $\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
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36 views

A map that is $(n-1)$-positive but not $n$-positive

Let $\phi : M_n(\mathbb{C})\to M_m(\mathbb{C})$ be a linear map. $\phi$ is called $k$-positive if the map $\phi^{(k)} : M_{kn}(\mathbb{C}) \to M_{km}(\mathbb{C})$, defined by evaluating $\phi$ ...
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44 views

Operator Graph Question

Let $T$ be closable. I am trying to show $\Gamma(\overline{T}) \subseteq \overline{\Gamma(T)}$. I can already show the reverse inclusion. Any ideas?
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2 positive decomposable maps

A positive map $\phi:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ is said to be $k$-positive if the natural extension ...
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81 views

The deficiency indices of symmetric operators

Given any pair of nonnegetive integer $(a,b)$, can you find an (unbounded) symmetric operator $T$ with the deficiency indices $(a,b)$? I guess the answer is yes, but how to do it?
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230 views

Integral operators with operator valued kernels

This is the definition for integral operators I know: Let $\Omega \subset \mathbb{R}^n$ and $D \subset \mathbb{R}^n$. Let $K : \Omega \times D \to \mathbb{C}$ be measurable. A linear operator $T: ...
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$\widehat{a}: \Omega(A)‎\rightarrow‎ \mathbb{C}~,~\tau‎ \mapsto \tau(A)‎‎ $

Suppose that $A$ is abelian Banach algebra for which the space $\Omega(A)$ is non-empty. If $a \in A$, we define the function $\widehat{a}‎‎$ by $$\widehat{a}: \Omega(A)‎\rightarrow‎ ...
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72 views

If limit of $f(n)$ is zero then the operator is compact

I want to prove the following: Suppose $\mathfrak{H}$ is the Hilbertspace $l^2(\Bbb{N})$ and $T_f$ the multiplication operator on $\mathfrak{H}$, thus $T_f\psi=f\psi$ for ...
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47 views

The hermitian element $h=\sum_{n=1}^\infty \frac{p_{n}}{3^{n}}$ generates $C_{0}(\Omega)$‎

‎Please help me to solve the following problem‎ : Let $\Omega$ be a locally compact Hausdorff space‎, ‎and suppose that the $C^{*}$-algebra $C_{0}(\Omega)$ is generated by a sequence of projections ...
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100 views

Exponential of an operator plus a constant term

I am reading a book on operator and matrix representation. Most of the examples are on Physics and they mention many terms like 'commute', i.e. the order of application of two operators might be ...
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103 views

Finding the spectral radius and spectrum .

I am solving the following question : If $k:[0,1]^2\to \mathbb C$ is continuous and $T_k : C[0,1] \to C[0,1]$ such that $$(T_kx)(t)=\int_0^t k(t,s)x(s) ds$$ Define $k_n: [0,1]^2\to \mathbb C$ ...
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30 views

How to show that density?

Show $$ \overline{\operatorname{span}(v_j)}=L^2([0,1]),~~~~~\overline{\operatorname{span}(u_j)}=L^2([0,1]) $$ with $$ v_j(x)=\sqrt{2}\cos((j-1/2)\pi x),~~~~~u_j(x)=\sqrt{2}\sin((j-1/2)\pi x). $$ ...
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84 views

Extension of differentiation operator to $L_2[0,1]$.

I'm studying for my functional analysis exam. We are required to know the proof of the following, but I cannot figure it out. Consider $L_2[0,1]$ with orthonormal basis $(e_n)_{n=-\infty}^\infty$ ...
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72 views

Containment of an element to an operator system

This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
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66 views

Positive maps on $\mathcal{B}(\mathcal{H})$ to itself

Let us consider the set of positive maps $\phi:\mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{H})$ ($\mathcal{H}$ is Hilbert space). Can we characterize all the maps which satisfies the ...
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117 views

Composition of positive maps

Let $\chi_A:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ be a completely positive (cp) map defined as $\chi_A(x)=AxA^*$, where $A\in\mathcal{B}(\mathbb{C}^n)$. Clearly any cp map ...
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267 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...
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47 views

Similarity orbit of compact operators

I am considering a problem connecting the spectra of compact operators to larger class of operators. Since spectra are invariant under similarity, I wonder whether there is a good reference on ...