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

2
votes
1answer
171 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
206 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 ...
1
vote
1answer
161 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
162 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
37 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
63 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
154 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
51 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
147 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
26 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
146 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
46 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
46 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
45 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
97 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
179 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 ...
2
votes
0answers
28 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
97 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
73 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 ...
2
votes
2answers
86 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
65 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
63 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
118 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
51 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
109 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
28 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
47 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
100 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
173 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
31 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
29 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
116 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
146 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 ...
0
votes
1answer
57 views

A question about download the recent paper. [closed]

I am interested in the journal about operator theory, such as Studia Math and Operators and Matrices. However, my college do not buy some journals. How can I get the paper from these journal?
1
vote
1answer
61 views

differentiability/holomorphicity of family of bounded operators

Edit: It seems I made a mistake in the statements on differentiability. I will replace weak differentiable implies strong differentiable with weak continuously differentiable implies strongly ...
2
votes
3answers
63 views

Operator attains norm

We have the following linear and well-defined mapping $T_{a}(b):=\sum_{j=1}^{\infty}a_{j}b_{j}$, with $\{a_{j}\}\in \ell^{1}$ and $\{b_{j}\} \in c_{0}$. Show that $\|T_{a}\|=\|a\|_{1}$. My work: ...
2
votes
2answers
57 views

What restrictions are there on explicit equations?

So I've always been told that for a function to be considered explicit it can only have one specific output for each input or simply pass the vertical line test. While I can accept that on it's face I ...
1
vote
1answer
51 views

Operators that are not represented as matrices , operating on matrices.

I am currently going through "Log-gases and random matrices" by PJ Forrester. I'm coming from a totally different academic background, and I cannot understand a point of his notation. More precisely, ...
4
votes
1answer
240 views

An alternate proof of Fuglede's theorem

To prove Fuglede's Theorem for normal operators on a separable Hilbert space, why does it suffice to show that $E(S_1)T E(S_2)=0$ for all disjoint Borel sets $S_1$ and $S_2$, where $E$ is the spectral ...
2
votes
1answer
55 views

Corollary to Putnam's theorem

Suppose $T_1$ and $T_2$ are normal operators on Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, respectively. Putnam showed that if $X$ is an operator satisfying $T_2X=XT_1$, then $T_2^*X=XT_1^*$. ...
4
votes
2answers
67 views

Given a vector x, can we say something about an A such that A x = x?

Let us assume that a vector $x \in \mathbb{R}^n$ is given and we are looking for a matrix $A \in \mathbb{R}^{n\times n}$ which yields $A x = x$. That is, we perform a sort of reverse questioning: ...
1
vote
0answers
98 views

Compactness of advection diffusion operator (elliptic operator)

Consider the ADE defined on a compact domain $\Omega \in \mathbb R^n$, with boundary $\partial \Omega$ Assume $u(x)$ is smooth incompressible vector field, with $u(x).\hat n(x)=0$ for $x\in \partial ...
1
vote
1answer
199 views

Self adjointness of square root operator

Theorem: If $A$ is self adjoint and nonnegative, then $A$ has a unique nonnegative square root $A^{\frac{1}{2}}$. As I understand, thesis of this theorem say only about the existence of ...
3
votes
1answer
74 views

Is $B - B'$ self-adjoint provided $B,B'$ are positive operators?

If I have two positive operators $B,B'$ on an arbitrary Hilbert space $\mathcal{H}$ not necessarily over $\mathbb{C}$, how do I know that $B - B'$ is self adjoint? EDIT: Reed and Simon define ...
3
votes
1answer
48 views

When does an operator commute with another operator given by a series?

Suppose $B$ is a bounded operator on some Hilbert space $\mathcal{H}$, given by a series of the form $$ B = I + \sum^\infty_{k = 1} c_k(I - A)^k $$ where $A$ is a given bounded operator on ...
6
votes
3answers
589 views

Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels

I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of ...
2
votes
3answers
2k views

Commutator of $x$ and $p^2$

I have a question: If I have to find the commutator $[x, p^2]$ (with $p= {h\over i}{d \over dx} $) the right answer is: $[x,p^2]=x p^2 - p^2x = x p^2 -pxp + pxp - p^2x = [x,p]p + p[x,p] = 2hip$ But ...
0
votes
1answer
19 views

Difference operator endomorphism

Let $\delta : R_{p}[x] \to R_{p}[X] $ the endomorphism of $R_{p}[X]$ such that : $\delta(P(X)) = P(X + 1) - P(X)$ , what is the kernel of $\delta$ ? (i tried to compute it explicitly but that was a ...
2
votes
6answers
408 views

Nilpotent linear operators

Suppose that $T : V \to V$ is a linear operator on an $n$-dimensional vector space $V$. (a) Show that for all $i$, $\ker T^i \subset \ker T^{i+1}$. (b) Show that if $\ker T^k = \ker ...