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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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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80 views

Wave Operators: Calculus

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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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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40 views

Strictly positive element in a C*-algebra

Searching about strictly positive elements, I found this exercise. I tried to solve it, and the following is my attempt. Please check my proof. Is it correct? Suppose $a$ is strictly positive. By ...
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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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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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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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Linear and nonlinear operator on normed space and its properties

My first question is : We know every bounded linear operator is continuous operator , and every continuous linear operator is bounded operator , in other words the continuousness and boundedness are ...
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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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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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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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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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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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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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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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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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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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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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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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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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(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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3answers
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Surjectivity of $Id-A$ for linear operator $A$ on Banach space with $\|A\|<1$

Let $X$ be Banach space and $A:X\rightarrow X$ linear opeartor such that $\|A\|<1$. It is clear that $Id-A$ is injective. Why is it also surjective?
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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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Proof involving strongly continuous semigroups.

Let $ (T(t))_{t \geq 0} $ be a $ C_{0} $-semigroup on a Hilbert space $ X $ with an infinitesimal generator $ A $, and let $ \rho \in (0,1) $. I want to prove that $ \displaystyle \sup_{t \geq 0} \| ...
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115 views

Compact Operators: Trace

Given a Hilbert space $\mathcal{H}$. Consider a bounded operator: $$A:\mathcal{H}\to\mathcal{H}:\quad\|A\|<\infty$$ Regard ONB's: ...
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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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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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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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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 ...
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Find the spectrum of the operator $T: \ell^2(\mathbb{C}) \to \ell^2(\mathbb{C})$ defined by $(Tx)_n = \frac{x_n}{n}$

Consider the linear operator $T:\ell^2(\mathbb{C}) \to \ell^2(\mathbb{C})$ defined as $$ (Tx)_n = \frac{x_n}{n}, \quad x \in \ell^2(\mathbb{C}). $$ I can show that it is bounded with norm $\|T\|=1$, ...
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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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Spectrum of a bilateral shift

Let $u$ be a bilateral shift on Hilbert space $\ell^2(\Bbb Z)$. As for unilateral shifts, the spectrum of $u$ does not contain any eigenvalue. Also $u$ is unitary, so $\sigma(u) \subset \Bbb S$ ($\Bbb ...
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Wot convergence and sot convergence

Let $\{A_n\} $ be a sequence of bounded linear operators on Hilbert space $H$ and $\langle A_n\xi,\eta \rangle \to \langle A \xi,\eta\rangle$ for $\xi,\eta\in H$ with $\|\eta\|=1$. Show that $\|A_n\xi ...
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Showing that an operator generates a unitary group

Consider the following operator on $X=L^{2}(0,1)$: $\displaystyle Af=\frac{df}{d\zeta}$ with domain: $D(A)=\{f\in L^{2}(0,1)|f$ is absolutely continuous, $\frac{df}{d\zeta}\in L^{2}(0,1)$ and ...
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The $C_0-$group generated by the operator $(Af)(x)=f'(x)+a(x)f(x)$

Consider the Banach space $L^1(\mathbb{R})$ of integrable functions $f:\mathbb{R}\to \mathbb{R}$. Consider the unbounded operator $A$ defined by $$(Af)(x)=f'(x)+a(x)f(x), \ \ \ x\in \mathbb{R}$$ for ...
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Quadratic Functional Differentiability

I would like to solve the following: Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. Consider the quadratic functional $\Phi$ defined by: \begin{equation} ...
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Binomial-like expansion for non-commuting operators

I would like to evaluate the following and express it in a compact form: $(\hat{a}^\dagger(x)+\hat{a}(x))^n\,\vert0\rangle$, where the $n^{\text{th}}$ power of the sum of the annihilation and creation ...
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How to calculate the adjoint of an operator and its domain?

Let $A : D(A) \subset L^2(0,1) \to L^2(0, 1)$, $$D(A) = \{u \in H^2([0, 1]) : u(0) = u'(1) = 0\}$$ $$Au = u''.$$ Can someone explain how to calculate the adjoint of A, $A^*$, and the domain of $A^*$, ...
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Showing that a domain of an operator is dense in $L^2$

Let $A : D(A) \subset L^2(\Omega) \to L^2(\Omega)$, where $$D(A) = \{u \in H^2([0,1]) : u(0) = u_x (1) = 0\}.$$ Show that $D(A)$ is dense in $L^2((0, 1))$. $D(A)$ is dense in $L^2((0, 1))$ if ...
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an operator question

I know how the derivative operator $\Big(\frac{d}{dx}\Big)^n$ works. But then how does it work if I have $$\exp{\Big(a\frac{d}{dx}+b\frac{d^2}{dx^2}\Big)}f(x)$$ I thought to use $$\exp ...
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50 views

Operator matrix is invertible if and only if its determinant is invertible

Let $A,B,C,D$ are pairwise commutative operators on a Hilbert space $H$, then a necessary and sufficient condition that the operator matrix $$\begin{pmatrix} A&B\\C&D\end{pmatrix}$$ be ...
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Does this show that it is a bounded linear operator?

Let $X$ be a Hilbert space and $A\in\mathcal{L}(X)$. I want to show that $\displaystyle e^{At}:=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}=T(t)$ defines a strongly continuous semigroup (i.e. a ...
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Differential of an operator $\phi: Mat_{2 \times 2}{\mathbb{R}} \rightarrow Mat_{2 \times 2}{\mathbb{R}}$

Let's consider an operator $ \phi: Mat_{2 \times 2}{\mathbb{R}} \rightarrow Mat_{2 \times 2}{\mathbb{R}}$ so that $A \rightarrow A^{-1}$. How to evaluate its differential? By the differential we ...
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The closed unit ball is not compact in infinite dimension spaces. Why?

We know that in finite dimension spaces the closed unit ball is compact, that is if H is a finite dimension space, then there exists an $u$ in the closed unit ball in H and $T \in \mathcal{L}(H, ...
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Calculating a norm of an operator

Let $T \in (C([a, b]))^*$, $$ T(u) = \underset{a}{\overset{(a+b)/2}\int} u(x) dx - \underset{(a+b)/2}{\overset{b}\int} u(x) dx. $$ Show that $ || T || = b - a $. We have that $$|| T || = ...
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Application of Uniform Bounded Principle (UBP)

Let $Y$ be a Banach space, and $Z$ be a n.v.s. If $(B_n)_n\in L(E,F)$ with the property that for all $(y_n)_n\in Y$, that $\|y_n\|\rightarrow 0$, we have $\|B_n(y_n)\|\rightarrow 0$. Prove that ...