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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Construct a unitary operator U on H

Given an infinite dimensional Hilbert space $H$. Let $|\lambda_k| = 1$ for $k = 1, ..., n$. Construct a unitary operator $U$ on $H$ such that $σ(U) = \{\lambda_k\}$ for $k=1,....,n.$ I can approach ...
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Construct a bounded linear operator S on H such that σ(S) = A

Given an infinite dimensional Hilbert space H. Let A⊂ℂ be closed and bounded. Construct a bounded linear operator S on H such that σ(S)=A.(σ(S) is spectrum of S) I can not find how to approach this ...
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10 views

Extened of a representation

The following is a part of a theorem of Folland's book: Let $X$ be a compact space, $B(X)$ the space of bounded Borel measurable functions on $X$, and $C(X)$ the space of continuous function on $X$. ...
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1answer
31 views

Show that a subspace is closed in Hilbert space $H$

Let $u\in B(H)$ , $\lambda < 0$. Also we have $\|(u-\lambda)x\|\geq |\lambda|\|x\|$. So $u-\lambda$ is bounded below. To show $(u-\lambda)(H)$ is closed in $H$, suppose $\{(u-\lambda)x_n\}$ be ...
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1answer
34 views

Show that an operator is well-defined

Let $v\in B(H)$, Define $u:|v|H\to H$ such that $u(|v|\xi) = v\xi$ . To show the map $u$ is well-defined, the author writes $$\||v|\xi\|^2=\langle v^*v\xi,\xi\rangle = \|v\xi\|^2$$ But I do not know ...
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Error in the calulation of the spectrum of the image of right shift operator in the Calkin algebra

If $S \in \mathcal{B}(\ell^2(\mathbb{N}))$ is the right shift operator $$ S(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots),$$ and $\mathcal{C} := \mathcal{B}(\ell^2(\mathbb{N}))/\mathcal{K}$ is the Calkin ...
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1answer
18 views

Partial isometry and projection

The following is a Theorem of Murphy's C*-algebras and operator theory: Let $H_1, H_2$ be Hilbert spaces and $u\in B(H_1,H_2)$. If $u^*u$ is a projection, then $uu^*u=u$. To show it, for $\xi\in ...
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2answers
44 views

Equality of two operators

The following is a fact in Murphy's C*-algebras and operator theory page 49: Suppose $u,v \in B(H)$, where $H$ is a Hilbert space, then $u=v$ if and only if $\langle u\xi,\xi\rangle = \langle ...
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1answer
15 views

operators on Hilbert spaces have adjoints

The following is a Theorem of Murphy's C*-algebras and operator theory: In the last line of proof, he claims $u^*$ is linear, but I think it's conjugate linear because for $y_2,x_2\in H_2$, $x_1\in ...
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2answers
21 views

Why does the set of an hermitian operator's eigenfunctions spans the functions space

During a discussion about linear hermitian operators, my professor claimed that if a linear operator $M$ is hermitian under a certian set of conditions, then genrally any function that fulfills these ...
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31 views

Commutativity and norms of specific operators (Problem 2.7.10 in Kreyszig's functional analysis book)

This is Problem 2.7.10 from Erwin Kreyszig's Introductory Functional Analysis with Applications. Let $C[0,1]$ denote the normed space of all (real- or complex-valued) functions defined and ...
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projections on a Hilbert space [on hold]

Let $p, q$ be projections on a Hilbert space $H$. Then the following conditions are equivalent: $p < q.$ $pq = p.$ $qp = p.$ $p(H) \subseteq q(H).$ $\|p( x )\| < \| q ( x ) \|$, $(x \in H)$ $q ...
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18 views

Density and Fredholmness

Let $X$ be a Banach Space and $Y$ a dense subset of $X$. An operator $T:X \to X$ is said to be Fredholm if it has closed range, $\dim \ker(T)<\infty$ and $\dim coker(T) < \infty$. Here is my ...
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27 views

Problem 8, Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 8 in the Problem Set following Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications: Show that the inverse $T^{-1} \colon R(T) \to X$ of a bounded ...
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Problem 2.7-9 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 9 in the Problem Set following Section 2.7 in the book Introductory Functional Analysis With Applications by Erwine Kryszeg: Let $C[0,1]$ denote the set of all (real- or ...
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1answer
69 views

If $H$ is a one-dimensional Hilbert space then the zero representation of a C*-algebra on $H$ is irreducible.

It says on page 143 of Murphy's $C^*$-algebras and operator theory that if $H$ is a one-dimensional Hilbert space then the zero representation of any C*-algebra on H is irreducible. What is the zero ...
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$s \in L^{1}(H)$ $\iff$ $s=\sum_{i=0}^\infty x_{i} \otimes y_{i} $

Let $H$ be a separable Hilbert space, and let $L^1(H)$ be the space of trace-class operators on $H$. I'd like to prove that $s\in L^{1}(H)$ if and only if there exists $\{ x_{i} \} , \{ y_{i} \} ...
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19 views

The resolvent of a differentation operator on $C[a,b]$

Consider a densely defined operator $A : C[a,b ]\rightarrow C[a,b ]$, $$Au=u^{\prime}$$ with domain $$D(A)= \{ u\in C^1[a,b]: u(b)=ku(a) \}$$ for some $k>0$. I have to find $R_A(\lambda)$ for ...
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1answer
28 views

The ideal generated by a non-compact operator

I wanted to find a quick proof of the following well-known fact. Since I couldn't easily find a reference, I provide a proof below. Let $H$ be a separable Hilbert space, and $J\subset B(H)$ be a ...
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32 views

Resolvent operator

Let's consider the following operator on $L^2(\mathbb{R}^3)$ $$A(t)=\Delta+b(t,x)\cdot\nabla$$ where $\Delta$ is the Laplace operator and $b(\cdot,\cdot)$ a smooth vector field. How to compute the ...
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If $I$ is a closed ideal in a C*-algebra $A$ and $J$ is a closed ideal in $I$ then $J$ is an ideal of $A$

The following is a remark of Murphy's C*-algebras and operator theory: . I do not know why he uses approximate unit. I think for $a\in A$ and $b\in J^+$, we have $b\in I$ and $b^{1/2}\in I$($I$ is ...
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62 views

Problem 2.7.6 in Kreyszig's Introductory Functional Analysis with Applications

Suppose that $X$ and $Y$ are two normed spaces over the same field ($\mathbb{R}$ or $\mathbb{C}$). Show that the range of a bounded linear operator $T \colon X \to Y$ need not be closed in $Y$. ...
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30 views

Can a Local Fractional Differential Operator exist?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$. The derivative of $f$ is defined pointwise, and we say that $f$ is differentiable if the derivative exists in each point. Higher order derivatives are ...
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1answer
29 views

Dynamics: Schwinger-Dyson-Expansion

Given a C*-algebra $\mathcal{A}$ Consider a free generator $\delta_0:\mathcal{D}_0\to\mathcal{A}$ with $\overline{\mathcal{D}_0}=\mathcal{A}$. Introduce a perturbation ...
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14 views

Resolvent set of derivative

i don't understand how to find spectrum and resolvent set $A : C[ab ]\rightarrow C[ab ]$ $D(A)= ( u\in C^1[ab]: u(b)=ku(a) )$ for some $k>0$ need to find $R_A(\lambda)$ and $\sigma (A)$. I can ...
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Extending $*$-isomorphisms between $*$-algebras to cross products.

Let $G$ be a discrete countable group and suppose I have two $G$-$C^*$-algebras $A$ and $B$ such that there exists a $G$-equivariant isometric $*$-isomorphism $\varphi \colon A \to B$. One can extend ...
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1answer
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$\phi(A^+) \subset B^+$ when $\phi: A\to B$ is an isometric linear map

Let $\phi: A\to B$ be an isometric linear map between unital C*-algebras $A$ and $B$ such that $\phi(a^*)=\phi(a)^* (a\in A)$ and $\phi(1)=1$. Show that $\phi(A^+) \subset B^+$. Clearly $A^+ = \{a^*a ...
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23 views

Norm of an operator and eigenvalues

I have $K\colon L^2(0,T) \to L^2(0,T)$ a Hilbert-Schmidt integral operator (and so $K$ is linear, bounded, compact and self-adjoint) and I have obtained its eigenvalues and eigenvectors. From them, I ...
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1answer
17 views

Show that hermitian element $h=\sum p_n/3^n$ generates $ C_0(\Omega)$

Let $\Omega$ be a locally compact Hausdorff space, and suppose that the C*-algebra $C_0(\Omega)$ is generated by a sequence of projections $(p_n)_{n=1}^{\infty}$. Show that the hermitian element ...
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Spectra of periodic Schrödinger equations

This question might be a little bit physics-related, but I kind of have a deep interest to ask this here, cause I would like to get an idea of the Mathematics behind the things I just covered in my ...
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1answer
45 views

Self-adjointness

In another thread it was claimed that the operator $O : \operatorname{dom}(O) \subset L^2(-1,1) \rightarrow L^2(-1,1)$ is self-adjoint. $$Of(x)= \frac{f(x)}{{1-x^2}}$$ It is obvious that $$\langle O ...
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Preserve self-adjoint properties

I was thinking about this problem recently: Let $T$ be a self-adjoint operator on $L^2((-1,1),d x)$. Now you define an operator $G$ by $G(f) := T(\frac{f}{(1-x^2)})$ with $\operatorname{dom}(G):=\{f ...
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How to show that the operator $T(\{x_n\})=\{n x_n\}$ has closed graph?

Consider the subspace $$D=\left\{x\in \ell^2 \ \big|\ \sum_{n\in\mathbb N} n^2 |x_n|^2<\infty\right\}$$ of $\ell^2$, and let $T:D\to\ell^2$ be defined by $T(\{x_n\})=\{n x_n\}$. I need ...
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Definition of well-defined for special case

I have a question about what well-defined means in a certain case. For an operator from $X$ to its dual $X^{*}$, say $A:X \rightarrow X^{*}$,why does the definition of $A$ being "well−defined" seem ...
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1answer
31 views

Vector question involving an operator!

So, here's the problem: An operator H capable of operating on vector x, is defined in terms of a given vector a by: H x=(a * x) where $*$ representes vector product Given that ...
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Definition of nonlinear bounded operator

Hi I am interested in confirming the definition of a bounded operator for a nonlinear operator. Let $X,Y$ be normed spaces. It is well known that a linear operator between $T:X \rightarrow Y$ is ...
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1answer
21 views

Holomorphic Functional Calculus vs Borel Functional Calculus

I am currently learning about different kinds of functional calculus and I was wondering if I could get something cleared up. The first type of functional calculus we learned about was holomorphic ...
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53 views

$C^*$-algebras, von Neumann algebras, unbounded operators and quantum mechanics in connection

I am studying the theory of $C^*$-algebras, von Neumann algebras and unbounded operators in courses on Functional Analysis and Opertor Algebras. Now I want to apply this knowledge to (algebraic) ...
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Strong derivative of a compound map

I find the strong Fréchet derivative of $\Phi(h,\psi(h))$, where $\Phi:T_0\times T_\xi\to Y$ with $T_0, T_\xi, Y$ Banach spaces and $\psi:T_0\to T_\xi$ is strongly differentiable in $0$, evaluated in ...
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1answer
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Closed restriction of an unbounded self-adjoint operator

Suppose $(A\,;\mathcal{D}(A))$ is an unbounded self-adjoint linear operator (obviously, $\mathcal{D}(A)$ must be dense) on a Hilbert space $\mathcal{H}$. Suppose $\mathcal{D}(C)$ is a proper dense ...
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1answer
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Iterating average

If $f$ is a continuous function $[0,1]\to \mathbb R$, we define a linear application $T$ as follows $$T(f)(x)=\begin{cases} f(0) & \mathrm{if }~ x=0 \\[0.2cm] \displaystyle ...
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Boundary conditions Legendre equation

I have Legendre's equation $$L(f)=\frac{1}{\sin(\theta)} \left(- \frac{d}{d\theta} \sin(\theta) \frac{df}{d \theta} \right)$$ Now I know that after substituting $\cos(\theta) =x$ we get a ...
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Point spectrum of a nonlinear operator on finite dimensional space

Given a nonlinear operator $T$ mapping $\mathbb R^n$ into itself, are there any known conditions on $T$ ensuring that the number of points in its point spectrum is upper bounded by the dimension $n$?
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1answer
63 views

Eigenvalues of Left Shift + Right Shift in $l_2([0,\infty))$

This question appeared on an old final exam and I am having difficulty completing it for practice. Let $S_r$ and $S_l$ be defined on the hilbert space $l_2[0,\infty)\to l_2[0,\infty)$ as the ...
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100 views

Square root of unbounded operator

Let $T: \operatorname{dom}(T) \subset H \rightarrow H$ be a positive self-adjoint unbounded operator, then I want to define a UNIQUE(!) operator $A$ such that $A^{*}A = T$. Actually, this construction ...
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60 views

Continuity of implicit function

I find a proof of the following theorem in A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа (pp. 492-493 here): Let $X,Y,Z$ be Banach spaces, $U$ a neighbourhood ...
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1answer
29 views

Finite rank approximation of bounded operators on Hilbert space

Let H be a (finite dimensional) Hilbert space. The approximation property states that every bounded operator from H to itself can be approximated by a sequence of finite rank operators. My question ...
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31 views

Spectrum of adjoint operator

Let $X$ be a hilbert space and $T\in L(X)$ Show that: (i) $\sigma_c(T^*)=(\sigma(T))^*$ (ii) $\sigma_r(T)=((\sigma_p(T^*))^*)$\ $\sigma_p(T)$ (i): $"\subset"$ Let $\lambda\in\sigma_c(T^*)$ then ...
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operator ranges of a closed linear operator

I have a closed linear operator T on a Hilbert space H. Also I have that range R(T^n) is closed for some n $\in$ N and for non-zero $\lambda$, R(T-$\lambda$I) $\cap$ R(T^n) is closed and ...
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1answer
38 views

CAR- & Weyl-Algebra: Uniqueness?

Given a Hilbert space: $\mathcal{h}$ Consider representations of the CAR-algebra: $\mathcal{A}_\text{CAR}^{(\prime)}(\mathcal{h})$ In Bratelli & Robinson it is stated the uniqueness: ...