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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Nonunital C*-Algebras: Morphism contractive?

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. Suppose it misses a unit $1\notin\mathcal{A}$. Consider a *-morphism $\pi:\mathcal{A}\to\mathcal{B}$. Then it is contractive: ...
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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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Construct a unitary operator U on H with prescribed spectrum

Given an infinite dimensional Hilbert space $H$. Let $|\lambda_k| = 1$ for $k = 1, ..., n$. Construct a unitary operator $U$ on $H$ such that $\sigma(U) = \{\lambda_k\}$ for $k=1,....,n.$ I can ...
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Nondegenerate representation

By the definition, we say a representation $(\pi,H)$ is nondegenerate if $cl[\pi(A)H ]= H$. Below I have two theorem, the first from Conway's Functional analysis and the second from Takesaki's ...
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*-isomorphism of a C*-algebra into an involutive Banach algebra is norm increasing

The following is a proposition of Takesaki's Operator theory: My question: How does he assume, considering the C*-subalgebra generated by k instead of $*$-Banach algebra B? Are we sure that the ...
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The positive element in a C*-algebra

The following is a theorem of Conway's Functional Analysis: for the proof ($c\to a$), I think we can say: for $\lambda\in \sigma(a)\subset \Bbb R$, there is a character $h:C(\sigma(a))\to\Bbb C$ ...
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Is the image of a $*$-homomorphism $\pi:\mathcal{A}\to\mathcal{B}$ closed if $\pi(1)\neq 1$?

Setting Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with unit $1\in\mathcal{A}$. Consider a morphism: $\pi:\mathcal{A}\to\mathcal{B}$ without $\pi[1]=1\in\mathcal{B}$. Especially, it is a ...
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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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64 views

Polynomial Ring: Root vs. Remainder

Framework: Consider a unital ring: $e\in R$ and a given polynomial: $p\in R[X]$ (Note that I do not require the ring to be an integral domain.) Problem: If it has a root then it factorizes: ...
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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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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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33 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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Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
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Core for an unbounded operator.

A symmetric operator $T$ is called essentially self-adjoint if its closure $T$ is self-adjoint. If $T$ is closed, a subset $D \subset D(T)$ is called a core for $T$ if $\overline {T\upharpoonleft D} ...
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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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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 $\xi\in ...
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$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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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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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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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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Why are integral and differential operators commutative?

For instance, let's assume a constant 3D surface over time $S$. $$ \frac{d}{dt}\iint_S \mathbf B \cdot \mathbf{ds} \quad=\quad \iint_S\frac{\partial \mathbf B}{\partial t}\cdot \mathbf{ds} $$ Why ...
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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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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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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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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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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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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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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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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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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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Spectral Measures: Subspace Decomposition

Attention This thread has been split into this one and: Spectral Measures: Subspace Characterization Problem Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. ...
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373 views

Norm of the sum of projection operators

Is it true that $$|| a R+b P||\leq\max \{|a|,|b|\},$$where $a$ and $b$ are complex numbers and $P,R$ are (orthogonal) projection operators on finite-dimensional closed subspaces of an ...
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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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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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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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$\phi(A^+) \subset B^+$ 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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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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314 views

Matrix form of the differential operator $\sum_{k=1}^N x^k\frac{d^k}{dx^k}$

The following differential operator: $P(x,N)=\sum_{k=1}^N x^k\frac{d^k}{dx^k}$ is defined in $x\in\left[-1,+1\right]$. Is it possible to find a matrix form of this operator vs. $N$? Because it's ...
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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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Spectral Measures: Lebesgue

Preface This thread deals with dominated convergence for functional calculus: $$f_n(\omega)\to f(\omega)\quad(\omega\in\Omega)\implies f_n(E)\to f(E)$$ Framework Given a Borel space $\Omega$ ...
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C*-Algebras: Group vs. Derivation

Given a C*-algebra $\mathcal{A}$. Consider a *-derivation $\delta$. Does it always generate a group: $$\tau(t)=e^{it\delta}$$ But a group of *-automorphisms is a contraction group: ...
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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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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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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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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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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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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: ...
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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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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 ...