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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norm of a nilpotent matrix

A proof I was reading used the claim that $||{N}||_2$ = 1 for a nilpotent matrix $N$. I tried to prove it, and have a couple of questions on it. First, my "proof": We know that there exists a basis ...
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Borel Functional Calculus Question

Let $T$ be a bounded operator and $A=\sigma(T)$ be its spectrum. Let $A^n \subset A$ be sequence of subsets s.t $A^n \rightarrow A$ (in compact open topology so $x\in A$ belongs to all but finetly ...
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20 views

Operator differentiability

I was wondering, what techniques can one use to prove that an operator (let's say acting on real analytic functions and taking values in a Banach space) is infinitely differentiable? I know that, for ...
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11 views

Polynomial generator

If we let $\alpha$ be a multiindex, can we generate any polynomial in $\eta$ with coefficients as multiples of $\kappa$ $$ D_z^{\alpha}\text{exp}(i(\kappa(z)-\kappa(x)-\kappa'(x)(z-x))\eta)|_{z=x} $$ ...
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18 views

When can we get discrete spectrum?

Suppose that $T$ is a densely defined closed operator on a separable Hilbert space $H$. Form $N = T^*T$. Assume further that $T$ has a finite dimensional kernel and satisfies the commutation relation ...
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15 views

About Antilinear (possibly Unbounded) Operators

Let $T$ be an unbounded anti-linear operator on a Hilbert Space. I would like to know if there is a natural or easy way to see existence of adjoint of $T$, closability of $T$(such as when $T^*$ is ...
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21 views

Doubt on eigenvalues of normal operators

I'm trying to understand the solution of the following problem: $T$ is a normal operator. If $T( v)=\lambda v$, then $T^*(v)=\bar\lambda v$: The solution is: I didn't understand why we ...
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22 views

Cauchy Schwarz inequality with an operator

The standard Cauchy-Schwarz inequality is given by, $|\langle\Phi|\Psi\rangle|^2\le\langle\Phi|\Phi\rangle\langle\Psi|\Psi\rangle$ But now I'm intressted in what happens to ...
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9 views

about analytic operator valued function

What's the definition of an analytic operator valued function ? Can we consider an analytic operator valued function as analytic function?
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8 views

Comparison of subsets of spectrum

Let $X$ be a Banach space. $A$ is a linear closed and densely defined operator and $S$ is a bounded invertible operator. I want compare $\sigma_{e,S}(\lambda S - A)$ to $\sigma_{e,S}(A)$. Here ...
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24 views

Saturated Monotone and Increasing Mappings

Let $A : \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a monotone mapping, i.e., $$ \left( A(x) - A(y) \right)^\top \left( x-y\right) \geq 0 $$ for all $x,y \in \mathbb{R}^n$. Let $B : \mathbb{R}^n ...
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26 views

Prove operator convergence

I have to proove that given $X$ a normed space and $Y$ a Banach space, if the sequence of bounden linear operators from $X$ to $Y$ $\{A_n\} \rightarrow A$ and the sequence $\{x_n\} \rightarrow x$ then ...
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31 views

Split Step Fourier Algorithm

Consider the NLSE (Nonlinear Schroedinger equation) that can be written as the following partial differential equation: $$ \frac{\partial{A}}{\partial{z}}=({\cal{L+N}})A\quad\quad(1) $$ where $A: ...
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25 views

Second Level Operators:

What would be an example of an Operator $$H$$ such that for any and all explicit functions U $$H[u] = I$$ where I is some other function However, for some other Operator W ex: [d/dx] ...
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16 views

about spectrum of the sum of two operators

It is well known that if $A$ and $B$ are commuting bounded operators on a Banach space then it is familiar that $$ \sigma(A+B)\subseteq\sigma(A)+\sigma(B)$$ I would to ask if I can find suffisant ...
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60 views

Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Euler–Lagrange ...
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29 views

Pseudo-monotone operators research paper question

Hi I just want to know if anyone can see how the result (2.34) is obtained in the following research paper http://caa.epfl.ch/publications/9-Boccardo-Dacorogna1984.pdf. Thanks, I know that it is a ...
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25 views

Confusion with proving that some subspace of a Banach-Space is closed

So here is my problem, I am trying to show that, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded map. Then $T(X)$ is closed if $Y/T(X)$ is of finite dimension. While ...
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41 views

Orthogonality Condition Eigen Functions of Sturm-Liouville Operator

I was wondering if anyone could help to derive the orthogonality condition $$\int^b_a y_n(x)y_m(x)w(x)dx=\delta_{nm}$$ of the normalised eigenfunctions (denoted by $y_n$ with eigenvalues $\lambda_n$) ...
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42 views

H P S class operators and their inequalities

First few definitions: $A \in I(K)$ iff $A$ is isomorphic to some member of $K$ $A \in S(K)$ iff $A$ is a subalgebra of some member of $K$ $A \in H(K)$ iff $A$ is a homomorphic image of some ...
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26 views

About inverse of an operator

Let $ X_{p}:= L_{p}([-a,a]\times[-1,1], dxdv), (a>0,\, 1\leq p<\infty)$ and the operator $$\left\lbrace \begin{array}{l} S :X_{p} \rightarrow X_{p} \\ \qquad \psi \mapsto ...
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23 views

Domains of operators defined by quadratic forms

Consider a separable Hilbert space $H$. Say we have two lower-bounded, densely defined quadratic forms $a$ and $b$ with respective domains $D[a],D[b] \subset H$ such that $D[b] \subset D[a]$ ...
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53 views

Self adjointness for functionals

I have posted this question already in the physics forum, but actually nobody could help. I am sorry, this question is related to quantum field theory. The Schrödinger equation of a free scalar field ...
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13 views

Operator's comparison

I have found the following two concepts: $\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$ and for any ...
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12 views

Fredholm operators and their applications

What are the applications (and possibly generalizations) of fredholm operators in partial differential equations?
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39 views

Largest/smallest cross-norm: A simple question about cross-norms on tensor products of Banach spaces.

This is a very simple dumb question as I’m completely new to the topic. I was reading Wikipedia’s entry on “Topological Tensor Products”, and there’s one thing I’m confused about. Let $ A $ and $ B $ ...
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29 views

Eigenvalues and eigenvectors of a nonlinear operator

I have found a few nice answers to the question: "Why are eigenvalues and eigenvectors useful." I can imagine that knowledge of eigenvectors (-values) for a general nonlinear operator is worthless. ...
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41 views

Does these two operators commute?

It is an exercise, $z\in\mathbb T$, which is the unit circle $$m(z)=\sum_{k\in\mathbb Z}a_kz^k$$ set $$Sf(z)=\frac{1}{\sqrt N} m(z)f(z^N)$$ So the adjoint operator is ...
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56 views

Show these operators converge to a particular limit

Let $H$ be a Hilbert space, and $T$ be a operator on $H$ of the form $T=\sum_{n=1}^{\infty}{\lambda}_{n}<x,e_{n}>e_{n}$ where $e_{n}$ are the eigenvectors of $T$ and an orthonormal basis of H ...
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28 views

Rotating the spectrum of a bounded operator

If $T$ is a bounded operator on a Banach space $X$, and $\sigma(T)$ is its spectrum, what would be an operator whose spectrum is $\sigma(T)$ rotated by $\theta$? For example, $-T$ has as spectrum ...
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Properties of solutions to an ODE

I have an ODE: $$ \frac{\mathrm{d}u}{\mathrm{d}t} + \mathcal{A}(t, u) = 0 $$ with final condition: $$ u(T)= \mathbf{1} $$ The function $u:\mathbb{R} \rightarrow \mathbb{R}^m$ is vectorial, and the ...
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26 views

Abstract Wiener space and integration related to a trace class operator

Suppose I have a trace class operator $A$ of a Hilbert space $H$. Also suppose I have an abstract Wiener space $(H,B)$. Then, $\langle Ax, x \rangle$ is defined almost everywhere in $B$ with respect ...
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22 views

Constants in Maximal regularity

We consider the following evolution equation \begin{equation} \left\{ \begin{array}{llc} v_t=A v+f,\\ v(0)=0. \end{array} \right. \end{equation} $A$ generates a bounded analytic semigroup on a Banach ...
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22 views

Hydrogenhamiltonian self-adjoint in one or two dimensions

let $d\in\{1,2\}$. I'd like to know if the operator $H=-\Delta - \frac{1}{|x|}$ is self-adjoint as an operator acting on a dense subset of $L^2(\mathbb R^d)$. In particular I'd like to know how its ...
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28 views

homomorphism question

Let B(H) be the set of bounded linear operators on a hilbert space H. Let F be a unital commutative subspace of B(H). Give an example of a homomorphism h from F to the complex numbers such that h is ...
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29 views

A couple of proofs on a spectrum

Let $T$ be a normal bounded operator. Let ${\lambda}$ be in $({\sigma}(T))$. Without invoking general algebra theories, show that: a) $p({\lambda},{\lambda}^*)$ is in $({\sigma}(T))$ for all ...
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Differential operator is self-adjoint

This is an exercise (13.8) in Rudin's Functional Analysis. Let an operator $T$ in $L^{2}(\mathbb R)$ be defined as follows: $\mathcal{D}(T)=\{f \textrm{ absolutely continuous}\in L^{2}(\mathbb{R}): ...
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31 views

induction on powers of a norm

Let T be a self adjoint operator on a hilbert space. I wish to prove by induction that $||T^n||$=$||T||^n$. I have proved it for n=1 and n=2. So assume it is true for some n Then, ...
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52 views

Spectral decomposition for normal compact operator

My book says $Tx=(\alpha x_{\alpha})$ where the $\alpha$ are eigenvalues the of T. But the image of an operator is not in general a sequence. Do they mean these are the scalars in the linear ...
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29 views

spectral projections

Let $A$ be a von Neumann algebra, and $T$ be an hermitian element of $A$. Show that the spectral projections of $ T $ belong to $A$. Proof: the spectral projections of $ T $ commute with every ...
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21 views

Question about operator algebra

I'm not certain about the rules of operator algebra, and I am wondering if these statements are equivalent $$\left(z^2\frac{d}{dz}-2z\right)\cdot\left(z^2\frac{d}{dz}-2z\right)=$$ ...
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33 views

orbits of a bounded linear operator

What interesting or strong results are there concerning orbits of an operator and invariant susbpaces (either in banach or hilbert space)? Obviously, I know that an operator T has an invariant ...
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16 views

partial differential operators

As we know, there exists a semigroup for partial differential operators $A = \sum_{i,j=1}^N D_i(a_{ij}(\cdot)D_j)$, see (Klaus-Jochen Engel, Rainer Nagel, one-parameter semigroups). My question is ...
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82 views

Gateaux Derivative.

Let $X$ be a Banach algebra. For $f\in X$, an operator $F_t$ is defined as \begin{equation} F_t(f)=\begin{cases} \Big\{\frac{f^t}{t(t-1)}, & t\neq 0,1; \\ \\ ...
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20 views

Is that operator positive-definite?

Let's consider the integral operator $\phi(x) = \int\limits^1_0\psi(y)\ln\Bigl(\Bigl|\frac{\sqrt{1-x^2}+\sqrt{1-y^2}}{\sqrt{1-x^2}-\sqrt{1-y^2}}\Bigr|\Bigr)\,dy$. How to check is this operator ...
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Properties of an additive mappings which preserves projections

Let $A$ and $B$ be two $C^{*}$-algebras and $\Phi:A\longrightarrow B$ be an additive map which satisfies $\Phi(0)=0$, $\Phi(I)=I$ and $\Phi$ preserves projections, (i.e, $\Phi(P)=Q$ where $Q$ is also ...
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74 views

Proving that two operators are equal

So I'm trying to prove that there is an equivalence between $\langle \psi\mid T\varphi\rangle=\langle\psi \mid S\varphi\rangle$ and $\langle\varphi \mid T\varphi\rangle=\langle\varphi \mid ...
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Characterization of central (modulo the radical) elements of a Banach algebra

Let $A$ be a Banach algebra and $Z(A)=\{a \in A:ax-xa \in $ Rad $ A \ \forall x \in A\}$ be the centre (modulo Rad$A$). Then TFAE: 1) $a \in Z(A)$ 2) $\exists M >0$ such that $\rho(a+x)\leq ...
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operator's domain

Let $X$ be a Banach space, $A$ is closed densely defined operator on $X$ and $B$ is a bounded operator on $X$. If we denote by $\mathcal{D}(A)$ the domain of $A$. I ask about the relation (i.e. ...
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18 views

What is the modulus of smoothness for Wigner-Ville Distribution?

I heard today a seminar where the speaker talked about General Shannon Sampling operators and its modulus of smoothness. I can only find this article about the modulus of smoothness. I think you can ...