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.

learn more… | top users | synonyms

0
votes
2answers
84 views

Multiplication operator with a function non-vanishing on the cantor set

Let $M_f$ be the multiplication operator, which acts on bounded functions $g$ on the unit interval as $g\mapsto fg$, with $f:[0,1]\rightarrow \mathbb{C}$ such that $f$ is nonzero only on the Cantor ...
0
votes
2answers
153 views

Limit of nth power of operator norm

I am given a compact operator $A$ which lives in a Banach algebra and whose spectral radius obeys $\rho(A)=\lim_{n\rightarrow\infty}||A^n||^{\frac{1}{n}}<1$. Now I want to prove that this implies ...
1
vote
1answer
123 views

Calculate the norm of this operator

$C[0,1]=\{ f : [0,1]\to [0,1], f$ continuous$\}$ $||f||_\infty=\max_{t\in [0,1]} |f(t)|$ $T:C[0,1]\to C[0,1]$ defined by $$(Tf)(t)=\int_0^1e^{s+t}f(s)ds$$ Find $||T||$ The usual way to do this ...
1
vote
0answers
30 views

Piecewise Continuous Fredholm Kernel

Suppose I have a ``Fredholm equation of the second kind" with kernel $$ K(x, y) = 1[x \geq u]k(x, y) $$ where k(x, y) is continuous. Let $f_n(x)$ be a smooth continuous approximation of $1[x \geq ...
2
votes
2answers
103 views

Show that the operator is bounded in $L_p$

Consider the operator $C$, acting on functions $f$ on the unit circle $S^1 = \left\{ z \in \mathbb C \mid |z| = 1 \right\}$ by the rule $$ (Cf)(z) = \frac{1}{2\pi i} ...
3
votes
1answer
74 views

equality of two operators…

Please help me with the following problem( give some hints or references): Let $X$ be a Banach space and $B(X)$ be the algebra of bounded linear operators on $X$. Suppose that $A$ and $B$ are two ...
0
votes
0answers
91 views

Exchangability of inner product and integral in bochner spaces

For the linear operator $e \in \mathcal{L}(V,V^{*})$, and sufficiently small $\delta s \in V := L^2(0,T,L^2(D))$ and $p \in V$ we have; $$ E[\langle e^{*}p, \delta s \rangle_{V}] = \langle E[e^{*} p ] ...
1
vote
2answers
62 views

Composing Translations and Reflections

Since $f(-x)$ is a reflection of $f(x)$ in the $y$-axis and $f(x+a)$ is a shift of $f(x)$ by $-a$ units, so for the longest time I've assumed $f(-(x+a))$ is a shift of $-a$ and then reflected in the ...
1
vote
2answers
72 views

rank of $A \otimes B$

For two matrices $A$ and $B$, what would be the rank of $A\otimes B$ as a matrix? Seems to me that $rank(A\otimes B) = rank(A)\cdot rank(B)$. But I don't see an elegant proof...
1
vote
1answer
114 views

Why is the set of compact operators closed in the space of all bounded operators between Banach spaces?

Let $X$ and $Y$ be Banach space. $B(X,Y)$ is the vector space of all bounded linear maps from $X$ to $Y$. Also, $K(X,Y)$ is the set of all compact operators from $X$ to $Y$. Why is $K(X ,Y)$ ...
2
votes
1answer
193 views

Holomorphic functional Calculus in Dunford and Schwartz

I am currently studying the spectral theory for bounded operators as described in the book "Linear Operators" by Dunford and Schwartz because I would like to obtain a better understanding of the ...
1
vote
0answers
43 views

Generator of a transport semigroup on the torus

Consider on the Banach space $C( \mathbb T;\mathbb R)$ of continuous functions on the 1-dimensional torus $\mathbb T:=\mathbb R/\mathbb Z$ (equipped with uniform norm) the operator $L$ given by $$ L ...
1
vote
2answers
40 views

Is each element of factor(von Neumann algebra) a linear combination of projection?

Let $A$ be a factor von Neumann algebra. Then every element of $A$ can be writeen as a finite linear combination of projections in $A$. Is it right? I know a little about von Neumann algebra, thanks a ...
3
votes
0answers
55 views

Matrix-valued functions with lacunary Fourier series

This question is motivated by investigation of the operator space structure of Hankel matrices (which is surely well-known to the experts). Consider a lacunary Hankel matrix, i.e. a matrix ...
0
votes
2answers
101 views

Norm space, linear operator exercise, help please!

$f \in L_2[a,b]$ $Uf(s):=\int_a^bk(s,t)f(t)dt$ $k(s,t):[a,b]^2\to R $ continuous. show 1) $U:L_2[a,b]\to L_2[a,b]$, in other words, $Uf(s)\in L_2[a,b] \quad \forall f$ 2) $U$ is linear and ...
3
votes
1answer
64 views

Orthonormal bases for Hilbert spaces

In Reed and Simon (Functional Analysis) Theorem II.6 states that, given an orthonormal basis $\{ x_\alpha \}_{\alpha \in A}$ (not necessarily countable)for a Hilbert space $H$, every $y \in H$ can be ...
2
votes
1answer
255 views

Learning roadmap for Non-commutative Geometry [closed]

I am interested in learning Non-commutative geometry and K-theory of operator algebras. Please suggest a learning roadmap for this subject. My present knowledge of Measure theory & Functional ...
6
votes
1answer
264 views

Weak* operator topology and finite rank operators

We will say that ${T_i}\subset B(X,Y^*)$ converges to $T$ in W*-operator topology if $T_i(x)\rightarrow T(x)$ in W*-topology of $Y^*$( $\forall y\in Y \langle T_i(x),y\rangle \rightarrow \langle ...
2
votes
1answer
231 views

Adjoint of resolvent of self-adjoint, densely-defined operator on a Hilbert space

Let $H$ be a Hilbert space, $T=T^*$ a densely-defined linear operator on $H$. Denote the resolvent set of $T$ as $\rho(T)=\{\lambda\in\mathbb{C}~|~T-\lambda$ has bounded, everywhere-defined inverse}, ...
5
votes
1answer
229 views

What is the relationship between spectral resolution and spectral measure?

In Kadison and Ringrose's book "FUNDAMENTALS OF THE THEORY OF OPERATOR ALGEBRAS", the author gives the following theorem. Theorem: If $A$ is a self-adjoint operator acting on a Hilbert space ...
1
vote
0answers
41 views

Composition of analytic functions is analytic in a general setting, and are they continuous?

Regarding the notion of analyticity discussed in this setting: A possible equivalence for holomorphicity I wonder if this is truly the correct definition (even though it is from Dunford-Schwarz) An ...
0
votes
1answer
68 views

If $X‎^{*}$ has Daugavet property, then X is likewise, but the converse is not valid.

I read in an article that If $X‎^{*}$ has Daugavet property, then X is likewise. I do search its proof in webs and "google scholar or book" but i dont find it. if you can prove it or give me a topic ...
1
vote
2answers
207 views

Matrix representation of a co-domain restriction of a linear operator

Consider the finite-dimensional linear operator: $\mathcal{A}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3},$ with $Ax=y,$ $A=\left[\begin{array}{ccc} 1 & 0 & 1\\ 1 & -2 & -1\\ 0 & 1 ...
1
vote
0answers
56 views

Is this operator closed?

Consider the linear operator $H$ with domain $D(H) = S(\mathbb R)\subset L^2(\mathbb R)$, where $S(\mathbb R)$ is Schwartz space, defined by \begin{align} H\psi(x) = -ix^3\frac{d\psi}{dx}(x) -i ...
3
votes
1answer
173 views

Confusion in Gelfand theorem in C*-algebra.

I am reading HX Lin's book, named "An introduction to the classification of amenable C*-algebras", I can not understand a corollary of Gelfand theorem(Corollary 1.3.6): If a is a normal element in a ...
0
votes
1answer
27 views

Unbounded family of isomorphisms

Suppose $(T_\alpha)$ is a unbounded in norm family of onto isomorphisms acting on a Banach space. Does it follow that the family $(T_\alpha x)$ is unbounded for any non-zero $x$?
2
votes
1answer
251 views

uniform convergence on compact subsets of the linear,continuous and uniformly bounded operators.

Let $X,Y$ be normed spaces. Let $T_j : X\to Y$ be a sequence of linear and continuous functions, such that $\lVert T_j\rVert\lt K$ $\forall j$. If $T_j$ converges pointwise to $T$, prove that $T$ is ...
0
votes
0answers
50 views

tight frame for $\mathbb{C}^N$

I have a question to ask Prove that if $K\in\mathbb{Z}-\{0\}$, then $\{\phi_p[n]=\exp(i2\pi pn/(KN))\}_{0\leq p<KN}$ is a tight frame of $\mathbb{C}^N$, i.e. $\sum_{k}|\langle f,\phi_p\rangle ...
2
votes
1answer
55 views

A question about compact Hausdorff space

Let $X$ be a compact Hausdorff space and $C(X)$ be the set of continuous functions on $X$. And $F$ is a closed subspace of $X$. If the $f\in C(X)$ such that $f|_{F}=0$ is only zero function( i.e. ...
1
vote
0answers
49 views

A question about bounded operators on banach space [duplicate]

Let $L(X)$ denotes the Banach algebra of all bounded linear operators acting on a Banach space $X$. And $T$ is not invertible. Can we find a invertilbe bounded operator series $\{T_{n}\}$ such that ...
2
votes
0answers
100 views

A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Jewett.pdf , Pg. 10, 2.3E ) it is said ...
0
votes
1answer
259 views

Example of non-orthogonal projection on Hilbert space

Can anybody cook up an example of a projection operator $P$ on a Hilbert space $H$ that is non-orthogonal? I.e., one where $PH$ and $(1-P)H$ are not orthogonal subspaces of $H$. I'm completely ...
3
votes
0answers
53 views

Uniqueness of solution to integral equation for operator valued functions

Let $X$ be a Banach space. Suppose I have a 2 parameter family of bounded operators on $X$: $V(t,s)$, $0\leq s\leq t \leq T$, such that $V(t,s)x=U(t,s)x+\int_s^t V(t,r)H(r)U(r,s)x\,dr$ and ...
1
vote
1answer
116 views

What is a concrete example of a non-compact Hermitian operator on an infinite-dimensional Hilbert space whose eigenvectors do not form a complete set?

If I am not misunderstanding anything: by the spectral theorem, Hermitian operators that act upon finite-dimensional Hilbert space as well as compact Hermitian operators that act upon ...
0
votes
1answer
86 views

bounding operator norm by quadratic forms on orthonormal basis

Let $\{u_i\}_{i=1}^n$ be a set of orthonormal basis and $M$ a symmetric matrix. Suppose \begin{equation} \left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n. \end{equation} Can I ...
3
votes
2answers
346 views

Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$?

I am a beginner of funcional analysis. I have a simple question when I study this subject. Let $L(X)$ denote the Banach algebra of all bounded linear operators on Banach space X, $T\in X$ is ...
1
vote
2answers
82 views

How can I better understand manipulating “operators” in mathematical relations?

Sometimes, (especially in physics), it's common to see mathematical relations manipulated and/or derived by separating "operators" from the things they "act on." I can usually keep up with and follow ...
1
vote
0answers
76 views

Self-adjointness of differentiation operator

Let's say $\mathcal{H}=L^2([0,1])$ and $p$ is the operator $-i\frac{d}{dx}$ defined on $\mathcal{D}(p)=\{f\in L^2([0,1])\ |\ f'\in L^2, f(1)=e^{i\theta}f(0)\}$. I have to prove that $p$ is ...
0
votes
2answers
64 views

if T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎

I want to show that if the operator T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎ Its proof is obvious from one hand.But i cant prove that ...
4
votes
1answer
132 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
3
votes
1answer
60 views

A bounded everywhere defined operator that is affiliated to a von Neumann algebra is in the algebra

A possibly unbounded operator $T$ on a Hilbert space $\mathcal H$ is (in my source) defined as affiliated to a von Neumann algebra $M$ if for each unitary element $u$ of $M^\prime$, $u^*Tu=T$ (or ...
1
vote
2answers
125 views

How would you determine whether this sequence transformation has an inverse?

Let $T : a \mapsto b$ be a transformation of sequence $a$ to $b$ of the form $$ T(a)_m = b_m = \sum_{k=1}^{\infty} a_k e^{-i 2 \pi m / k } $$ Question. How would you go about determining if this ...
2
votes
0answers
36 views

Pointwise convergence of resolvent

Suppose $T$ is a quasiniplotent operator and $\lambda_n$ a sequnce converging to $0$. Then clearly $||(\lambda_nI-T)^{-1}||\to\infty$. I am interested for which $x$ we have that ...
1
vote
1answer
48 views

Conditions necessary for commutators [A,B]=[B,A]?

I know that normally for commutators that [A,B]=-[B,A] where A and B are operators. But under what conditions does [A,B]=[B,A]?
4
votes
0answers
112 views

For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...
0
votes
1answer
265 views

Question about convergence in weak operator topology (from Reed and Simon)

I am reading over Chapter VI in Simon and Reed's Functional Analysis. In the first section, the discussion covers various topologies defined on $\mathcal{L}(X,Y)$, the space of bounded linear ...
0
votes
1answer
35 views

Definition of a norm infinity

I have $u:\mathbb{R}^3\times(0,\infty)\longrightarrow\mathbb{R}$ and $g:\mathbb{R}^3\longrightarrow\mathbb{R}$. Which means the following?: $\|u(\cdot,t)\|_{L^\infty(\mathbb{R}^3)}$, and ...
2
votes
1answer
36 views

Condition on spectrum of T

Let $T$ $\in \mathfrak{B}(\mathbb{H})$ be normal. Let $A$ be the closed subalgebra generated by $T$, $T^{*}$ and $I$. Suppose $T$ can be approximated in norm by finite linear combinations of ...
2
votes
0answers
139 views

Does ternary operations have associative property?

Binary Operation is a function. Right? We know that all Binary operations have associative property. They must be either associative or non-associative. The condition is : $$(a*b)*c = a*(b*c)$$ ...
1
vote
0answers
195 views

Comparison of Strong OPerator and Weak * Topologies on B(H)

It is known that in $\mathfrak{B}(\mathbb{H})$, the weak operator topology (WOT) is contained in both the strong operator topology (SOT) and $\sigma$-weak topology. In general the SOT and the ...