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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Modulus: Invariant Domain

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$M:\mathcal{H}\to\mathcal{K}:\quad \|M\|=1$$ Regard dense subspaces: ...
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Determining spectral bounds variationally.

I'm learning C0-semigroup theory (mainly from Arendt et al. (vector-valued Laplace-transforms and Cauchy problems), Engel & Nagel (One par. semigroups for linear evolution eq.),Evans (partial ...
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Some question about extension of bounded linear operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$H: ...
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how to solve Dirac Equation numerically?

The effective Hamiltonian for my system is: \begin{equation} H=\nu_{F} {\bf \sigma}\cdot\left(q-By\hat x\right) \end{equation} where ${\bf \sigma}$ and $q$ are the Pauli matrices and the momentum ...
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Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES. And in the paper, they provide an inequality of the Schatten-p (quasi-)norm, ...
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Selfadjoint Operators: Relative Boundedness

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard an operator: ...
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Approximate unit for an ideal and its limit

Let $A$ be a C*-algebra and $I$ is a closed ideal of $A$. If $(\pi, H)$ is a cyclic representation of $A$, Could we show $$\pi(u_i)\xi \to \xi $$ where $\{u_i\}$ is an approximate unit for $I$ and ...
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Møller Operators: Functional Calculus

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider Hamiltonians: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ $$K:\mathcal{D}(K)\to\mathcal{K}:\quad K=K^*$$ and a bounded ...
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The linearness of extension of linear bounded operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$G: ...
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Commutative Operators from QM

In Theoretical Chemistry, there seems to be a lot of assumptions about mathematics that are incorporated without justification. One example that I found questionable is this: $$\int \Psi_1^*\ ...
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Prove operator is isometry

Let $(X,\mathcal{A},m,T)$ be a probability preserving transformation. Prove that the operator $U:f\mapsto f\circ T$ satisfies $$ \|Uf\|_{p}=\|f\|_{p} $$ for every $1\le p<\infty$. My idea: $$ ...
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What happend if the divergence of a vector field is zero?

I just want to be sure if I'm wrong or not, I want to know what happend for a vector fiel if his divergence is zero ? Are the vectors have all the same lengh ? Or maybe are they all time parallel ? ...
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Continuous Linear Operator in $\mathbb{R}$- normed spaces.

Let $E$ and $F$ $\mathbb{R}$-normed spaces and let $f:E\longrightarrow F$ satisfying: $f(x+y)=f(x)+f(y) \,\forall x,y\in E$; $f$ is bounded in the unit ball $B_E = \{x\in E: \Vert x\Vert ...
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Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, ...
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Self-adjointness under relatively bounded perturbation

Let $T$ be a densely defined linear operator on a Banach space $X$. Another operator $A$ satisfying $\mathcal{D}(T) \subset \mathcal{D}(A)$ is called a relatively bounded perturbation of $T$ if ...
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Spectral projection and isolated point of spectrum

Let $u\in B(H)$ be a normal element with spectral resolution of the identity $E$ and $\lambda$ be an isolated point of spectrum $u$. Show that $E(\lambda)H = \ker(u-\lambda)$ . I can show that ...
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Showing that if any non-zero $f \in X^*$ takes its maximum value on the unit sphere at most once, then X is strictly convex

Let $(X, \| \|)$ be a normed space. I'm trying to show that if all non-zero $f \in X^*$ take their maximum value on the unit sphere at most once, i.e. $\forall f \in X^* - \{0\}$ there is at most one ...
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On closed ranges and sequences which converge to zero

I'm reading a proof of the Fredholm alternative, and there is a claim that goes like this: Let $K:X\rightarrow X$ be a compact linear map. Define $T=I-K$, then $Y=\ker(T)$ is a finite dimensional ...
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Problem 8 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein and Shakarchi's Real Analysis

The following is problem 8 from chapter 4 ("Hilbert Spaces: An Introduction") of Stein and Shakarchi's Real Analysis. Suppose $\{t_k\}$ is a collection of bounded operators on a Hilbert space $H$. ...
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Are “most” operators on an infinite-dimensional complex Banach space “diagonalizable”? [closed]

This is true for finite-dimensional spaces, of course. To be precise, let $T$ be an operator on a complex Banach space $X$ which is not finite-dimensional. For each $\lambda \in \mathbb{C}$, let ...
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Spectrum of periodic schrödinger operators

In many articles it's stated, as if it's common knowledge, that any Schrödinger operator with periodic potenial has purely absolutely continuous spectrum. I've tried to actually find a theorem ...
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Distance between a density operator and a pure quantum state.

Given density operators $\rho_1$ and $\rho_2$ and a pure quantum state $|\psi>$. It is promised that $|\psi>$ is in only one of the given density operators. How to find which density operator ...
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Norm of Operator Proof

I'm stuck on this problem that I can't seem to figure out. Here's the problem. To note, equation 2.42 says that $$||T|| = \sup \{ ||Tu||: u \in C([a,b]), ||u|| = 1 \}$$ where $T$ is defined, ...
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Partial Isometries: Subspaces

Note: This thread is not to gain reputation!!! Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a bounded operator: $$W:\mathcal{H}\to\mathcal{K}:\quad\|W\|<\infty$$ Then a partial ...
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States: Density

Problem Given a Hilbert space $\mathcal{H}$. Regard the CAR-algebra: $$\{a(\eta),a(\zeta)\}=0\quad\{a(\eta),a(\zeta)^*\}=\langle\eta,\zeta\rangle$$ Consider a density: ...
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36 views

spectral theory expandable to arbitrary polynomials?

Given a Banach space $X$ and closed operators $A_i$ ($i \in \left\{0,...,n\right\}$) which have a common domain $D$ that is dense in $X$. An obvious candidate for the title of "generalised resolvent ...
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Distance preserving function on a Hilbert space

Let $\Bbb F = \Bbb R$. Show that every preserving function $f$ on Hilbert space $H$ has the form $f(x) = f(0) + Tx$ for some isometry $T$ in $B(H)$. If $f$ is linear then $f$ is an isometry. Suppose ...
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Example of an ideal in $C(\Bbb D)$ that is not self adjoint

Give an example of an ideal in the C*-algebra $C(\Bbb D)$ that is not self adjoint. My attempt: The function $f: \Bbb D \to C$ such that $f(t) := t+i$ belongs to $C(\Bbb D)$. Let I be the ideal ...
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(Updated) Geometric Illustration of Monotone and Maximal Monotone Maps

I am writing a note about the Monotone and Maximal Monotone maps from the following book http://link.springer.com/book/10.1007%2Fb97594 In this book we read a map ...
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restriction of irreducible representation to an ideal is irreducible

Let $A$ be a C*-algebra and $I$ a closed left ideal of $A$. Show that if $\{\pi,H\}$ is an irreducible representation of $A$, then the restriction of $\pi$ to $I$ is either zero representation or ...
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Domain of square root of a self-adjoint positive operator

Let $A \geq 0$ be a densely defined self-adjoint positive operator on a Hilbert space $H$ obtained by Friedrichs extension, and let $Q$ be the densely defined quadratic form associated to $A$, that ...
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Sobolev spaces on compact manifolds

Let us consider a self-adjoint elliptic pseudodifferential operator $P \in OPS^2$ on a compact manifold $M$ such that $spec(P) \subset (0, \infty)$. Is the norm $(Pu, u)^{1/2}$ on $H^1(M)$ equivalent ...
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Inversion of differential operator

My goal is to solve the differential equation, written in the following form $$\Big(\frac{d}{dx}+I\Big)^{2n}V(x)=x+C$$ where $C$ is some constanst. I want to do it by the operator method. Namely one ...
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Fractional Laplacian on the torus [duplicate]

Consider the Laplacian on the $n$ dimensional torus $T$, given by $-\Delta : L^2 \rightarrow L^2$. Let the domain of $-\Delta$ be all $C^\infty$ functions initially. Now consider the Friedrichs ...
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Fractional Laplacian on the torus

Consider the Laplacian on the $n$ dimensional torus $T$, given by $-\Delta : L^2 \rightarrow L^2$. Let the domain of $-\Delta$ be all $C^\infty$ functions initially. Now consider the Friedrichs ...
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Mapping properties of differential operators: Reference for targeted reading

In my studies (currently I am trying to understand spectral properties of differential operators) I am encountering operators that are unbounded. To be more concrete, here is an example that I ...
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When open mapping theorem fails.

Let Y = $L^1 $($\mu$) where $\mu$ is counting measure on N. Let X = {$f$ $\in$ Y : $\sum_{n=1}^{\infty}$ n|$f(n)$| Define T : X -> Y by $Tf(n)=nf(n)$ Now Let $S=T$$^{-1}$ Show $S$ is not open. ...
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Notation in Reed/Simon Vol. IV (and possibly an earlier volume)

I'm wondering if there are any mathematical physicists/analysts out there that can help me with some notation I've seen in Reed and Simon's books on analysis. Unfortunately I don't have time to read ...
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$A\subseteq B(X, Y)$ compact if and only if closed and $Ax$ is conditionally compact

This comes from Exercise 2 of Chapter VI in Dunford & Schwartz. I am trying to prove the following statement: A set $A\subseteq \mathscr{B}(X, Y)$ is compact in the strong operator topology if ...
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Compact Operators: Separable Range

Given Banach spaces $E$ and $F$. Consider a bounded operator: $$T:E\to F:\quad\|T\|<\infty$$ Certainly one has: $$T\text{ compact}\implies\mathcal{R}T\text{ separable}$$ What about the converse?
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Compact Operators: Trace

Problem Given a Hilbert space $\mathcal{H}$. Consider a bounded operator: $$A:\mathcal{H}\to\mathcal{H}:\quad\|A\|<\infty$$ Regard orthonormal bases: ...
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Showing that A is NOT an infinitesimal generator

As a state space, choose $X=L^{2}(0,1)$. Let $A$ be defined as $\displaystyle Af=\frac{df}{d\zeta}$ with domain $D(A)=\{f\in L^{2}(0,1)|f$ is absolutely continuous and $\frac{df}{d\zeta}\in ...
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Polar Decomposition: Ranges

This is just a note. Given Hilbert spaces $\mathcal{H}$, $\mathcal{K}$. Consider a closed operator: $$T:\mathcal{D}(T)\to\mathcal{K}:\quad ...
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is the complexification of a finitely strictly singular operator itself FSS?

Let $X$ and $Y$ be real Banach spaces, and let $X_\mathbb{C}$ and $Y_\mathbb{C}$ denote their respective complexifications. Suppose $T:X\to Y$ is a bounded linear operator which is finitely strictly ...
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Finding the infinitesimal generation of a strongly continuous semigroup

Let $X$ be a Hilbert space, $A\in\mathcal{L}(X)$ and $\displaystyle T(t)=e^{At}=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}$. I have already shown that $T(t)$ defines a $C_{0}$ semigroup. But now I need ...
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wot limit of a sequence of projections

Let $\{P_i\}$ be a net of projections on a Hilbert space , then we can show wot limit of this net is a projection, too. I saw below example of a sequence of projections which its wot limit is not a ...
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What is the spectral radius of the operator $T_k:C[0,1]\to C[0,1]$ defined as $T_k x (t)= \int_0^1 k(t,s) x(s) ds$?

I would like to know the spectral radius of the operator $T_k$ from $C[0,1] \to C[0,1]$ : $$T_k x (t)= \int_0^1 k(t,s) x(s) ds$$ where $k(x,y)\colon [0,1]^2 \to \mathbb C$ is continuous. And ...
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Compute the spectrum of the integral operator $K:L^2([0,1]) \to L^2([0,1])$ defined as $(Kx)(t) = \int_0^t x(s) ds$

Let $K:L^2([0,1])\rightarrow L^2([0,1])$ be the linear operator defined by $$(Kx)(t)=\int_0^tx(s)ds, \quad x \in L^2([0,1]).$$ Now I have to compute the spectrum, but I don't have any idea how to do ...
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Commutant of algebra of multiplication operators

Let $L^2(X)$ be the set of Lebesgue square-integrable functions on a locally compact Hausdorff space $X$. Define $\mathfrak{A}:=\{M_f:f\in L^{\infty}(X), f=\overline{f}\}$, where $M_f$ is the the ...