Science topics: AnalysisOperator Theory

Science topic

# Operator Theory - Science topic

Explore the latest questions and answers in Operator Theory, and find Operator Theory experts.

Questions related to Operator Theory

One of the main problems of semigroup theory for linear operators is to decide whether a concrete operator is the generator of a semigroup and how this semigroup is represented.

One idea is to write complicated operators, as a sum of simple operators. For this reason, perturbation theory has become one of the most important topics in semigroup theory. My question is about the multi-perturbed semigroups or multiple perturbation of semigroups in a Banach space. I need a recurrent formula for a semigroup perturbed by multiple (several, i.e. more than two) bounded (in general unbounded) linear operators. I have searched for it, but only found a simple case, called the Dyson-Phillips series for a semigroup generated by A

_{0}+A_{1}. How can we find the generalisation of this formula for a semigroup generated by A_{0}+A_{1}+...+A_{n}for a fixed natural n? Many thanks in advance. I am looking forward to your suggestions and recommendations on this topic.Let $A$ be a densely defined symmetric (unclosed) operator and let $B\in B(H)$ be positive.

I know that if $\overline{A}B$ is normal, then $AB$ need not be normal.

My question is: If $AB$ is normal, is it necessary that $\overline{A}B$ remains normal? (notice that, thanks to the normality of $AB$, I have shown that $B\overline{A}\subset \overline{A}B$).

Many thanks,

Hichem

Is $TT^*$ (or $T^*T$) densely defined if $T$ is a densely defined and symmetric linear operator?

I feel this is untrue, but do you have a counterexample?

Thanks,

Hichem

Let $B\in B(H)$ be self-adjoint and let $A$ be a densely defined symmetric (and closed if needed) operator such that $A^2$ is densely defined. If $BA^2\subset A^2B$ say, is there a result which gives $BA\subset AB$? We may add positivity to $A$ to avoid trivialities.

Notice that I already have a counterexample when $A^2$ is not densely defined.

Cheers,

Hichem

A quantum mechanical operator acting on the abstract state space of Dirac kets maps vectors of the space to other vectors of the space.

Assuming for simplicity that the examined quantum system is one dimensional, in position space, the same operator is a function of momentum, which is a differential operator, and position, which is a variable.

If the expression, or representation, of the operator in position space has singular points; i.e. values of the position where the operator is not defined for some reason; for instance due to a pole, is there a property of the same operator in state space that distinguishes it from operators that do not have singularities in position space?

Τhe position space is realized by using the position eigenstates which do not belong to the state space since they are not normalizable, but since the state and position spaces are physically (and mathematically) equivalent, shouldn’t a property exist that distinguishes operators with singularities from operators without singularities in position space?

Dear colleagues. The text of the question is available in the attached image or pdf-file.

Sincerely, Yaroslav Grushka.

As we know, the vastness of the subject is realized by the variety of interdisciplinary subjects that belong to several mathematical domains such as classical analysis, differential and integral equations, functional analysis, operator theory, topology and algebraic topology; as well as other subjects such as Economics, Commerce etc.

Recently, some new operators having the keyword "fractional" are proposed to define the non-integer order derivative.

For integer order derivatives, it is well known that the locality and validity of Libeniz rule are the main properties of integer order derivatives, but what is the characteristic property of a fractional derivative? When we can call an operator is "fractional" and what features should an operator have to call it a "derivative" operator?

If B(H) is the algebra of bounded linear operators acting on an infinite dimensional complex Hilbert space, then which elements of B(H) that can't be written as a linear combination of orthogonal projections ?

My question concersns operators on Hilbert spaces.

It is well-known that any family of mutually commuting compact self-adjoint operators can be simultaneously diagonalized. Is the same true for any family of mutually commuting bounded self-adjoint operators?

Let X be a Banach space and let T be a bounded linear operator on X. We know that if X is reflexive and T is compact then there exists x in the unit sphere of X such that T attains its norm at x. Can we impose any condition on X or on T such that x is the unique such point?

A most interesting project! There is still a marked need for work to identify fundamental first principles in biology!

From a modelling perspective you possibly work from objects (and their structure) towards interactions between objects (and their equations).

If so, the recently developed Operator Theory may offer guidance by offering a set of standard objects, and by unraveling natural organization/complexity along three dimensions: operators of different complexity, organization inside an operator, and organizations consisting of many operators. (Classical approaches to hierarchy almost all suggest a single linear dimension).

Look forward hearing more from you,

Gerard Jagers op Akkerhuis

U

*nder what conditions, a linear operator can have closed extensions?**Let T be a closable linear operator. Are there any others closed extensions of T except the natural extension $\overline{T}$ ? if "yes" are there any formal constructions of these extensions?*

let A be an algebra and A1 the unitization of A, then multiplication algebra which is denoted by M(A) is algebra generated by identity operator and left multiplication and right multiplication also for the algebra A1, we have M(A1)

it is clearly that M(A) is subset of M(A1) when we have M(A)=M(A1)?

Let $X$ be a complex Banach space and $T$ be a bounded operator acting on $X$. Let $\sigma(T)$ and $\sigma_{ap}(T)$ denote the spectrum and approximate point spectrum of $T$, respectively. Let $\lambda \in \sigma(T)$. It is classical that, with the aid of the spectral projection, $\lambda$ is isolated in $\sigma(T)$ if and only if $T - \lambda$ can be decomposed as a direct sum of a quasinilpotent operator and a invertible one. Now my nature question is as follows: Let $\lambda \in \sigma_{ap}(T)$. Is it true that $\lambda$ is isolated in $\sigma_{ap}(T)$ if and only if $T - \lambda$ can be decomposed as a direct sum of a quasinilpotent operator and a bounded belowness one? Here we say that an operator is bounded below if it is injective and its range is closed. It is also nature to find the answer for other spectra (eg. left spectrum, surjective spectrum, right spectrum, essential spectrum,...). Thank you!

I learn that in a Banach algebra over F (R or C), an element a is positive if and only if a=b*b, and equivalently a is positive if a = a* and spectrum of a is a subset of R of non negative elements ([0, +\infny). My questions are:

1) If F=R, that is real Banach algebra, what can we say about it's positive elements?

2) Do we still need involution on the real Banach algebra to have a positive elements?

3) Could you please give an example to justify the answer to Q2.

By adding to the title and its anisotropy of this project the non-local or (singular) integral operators = my response to an ask of feedback to this paper, see below (paper put in reference to my own project)= (+ -via google translation, French text below) it is quite interesting = the anisotropy is here xi-cartesian constant (and the study comes out of the 1d to speak of the true multidimensional) and one can imagine more without difficulties The localized varying declinations of this anisotropy with Cartesian bases and ellipsoids/paraboloids and weights and functions, and your localized exponents ai, varying in x all of them in their own way, in centers, angles and Rotations, values etc, for your operators and various weights and functions on which your operators apply, variabilities more or less slow, fast, (ir /) regular, adapted, local, nice or not etc. It is quite speaking when the function is the derivative or the gradient of another or with studies in spaces of besov, sobolev or all functional spaces Es,p,q, with s not zero, s being the index (Integral) of derivation (instead of the spaces with s = 0, Lp, Lq etc) with s, p and p variables in x = we approach geometric anisotropy, foliations and associated irregularities which can eg to account for very natural situations in mathematical physics such as vortex patches and many others. (+-via google traduction, texte francais plus bas) c'est tout a fait interessant = l'anisotropie est ici xi-cartesienne constante (et l'etude sort du 1d pour parler du vrai multidimensionnel) et on peut imaginer de plus sans trop de difficultés, les declinaisons variables localisées de cette anisotropie avec des bases cartesiennes et des ellipsoïdes/paraboloides et des poids et fonctions et vos exposents 'ai', tous localisés, variant en x tous chacun d'eux a sa facon, en centres, angles et en rotations, valeurs etc, pour vos operateurs et poids et fonctions diverses sur lesquels s'appliquent vos operateurs, variabilités plus ou moins lentes, rapides, (ir/)regulieres, adaptées, locales, gentilles ou pas etc. c'est assez parlant quand la fonction est la derivée ou le gradient d'une autre ou avec des etudes dans des espaces de besov, de sobolev ou tous espaces fonctionnels Es,p,q, avec s pas nul, s etant l'indice (integral) de derivation (au lieu des espaces a s=0, Lp, Lq etc) avec donc s, p et p variables en x = on s'approche de l'anisotropie geometrique, les feuilletages et foliations et des irregularités associées qui peuvent eg bien rendre compte de situations tres naturelles en physique mathematique comme les vortex patches et beaucoup d'autres.

(paper and project "Weighted Anisotropic Morrey Spaces Estimates for Anisotropic Maximal Operators" and "weighted anisotropic Morrey spaces...." by Ferit Gürbüz= I pronounce my self on its/their subject and not on the novelty that its author brings to it). the rg-profile of the author has 2 (slightly different) papers with same title, but one of them has a "full text" with the title ending with: "...AND 0 -ORDER ANISOTROPIC PSEUDO-DIFFERENTIAL OPERATORS WITH SMOOTH SYMBOLS" with an additional math paragraph.

Does there exists a function $g \in L^1(R)$ such that it satisfy the following property

(1)$ L^1$ norm of g \leq 1

(2) $\hat{g}(s) =1$ for $s\in [-\delta, \delta]$ and

(3) \hat{g} has compact support ?

More precisely

Denote $R=\mathbb{R}$, $S(R)= Schwarz Class function on R $

I know that $C_{c}^{\infty}(R) \subset S(R)\subset L^1(R)$ and Fourier transform is bijection on S(R). So one can start with a function with property $(2)$ and $(3)$ above and take $g$ as a inverse Fourier transform of the function. But I don't know that norm of g can be bounded by 1 ?

positive definite = positive and invertible.

operators are defined on a separable complex Hilbert space.

Since, classically we define force as rate of change of momentum and by weyl rule we can map classical functions to quantum operators. Then, why is it that we don't often hear the word 'force operator' in quantum mechanics?

Consider the normed linear space X=R^2 endowed with the usual l_p norm, p>1 and p other than 2, infinity. Consider the family of linear operators T_k (x,y)=(y, (x+y)/k), k>0. Is it true that each T_k attains their respective norms at only one pair of points of S_X? (The points would depend on k, of course!) In particular, I would like to especially know the answers for k= 1,2.

Please can anyone explain how the Bernstein durrmeyer operator becomes a contraction?

Hi, I have a linear bounded operator

*A*between tow Hilbert spaces with a dense domain and dense range, is it surjective? If the answer is no, are there more conditions on*A*to be surjective?Before, I did some work about the queuing model by using functional analysing method(operator theory), it was related to abstract Cauchy problem. Now ,can we discuss the abstract Cauchy problem by changing it to convex programming? is it possible?

Are there any unitary operators on the Bergman space $A^2(D)$?

Let M be a convex and compact subset of m by n matrices (m is strictly less than n). If for each A in M, A has full rank, then there is an d in R

^{n}such that for each A in M, the matrix [A^{T}d^{T}] has full rank.I was wondering if someone proves this conjecture or provides a counterexample.

Any suggestion by using the finite element method?

There are unitary, self adjoint and normal operators in operator theory. Do each collection posses nice property?

For what type of separable C* algebras, there is no any integer n and any nontrivial morphism from A to the Cuntz algebra O_{n}? In the other word for what type of C* algebras, Hom(A, O_{n}) is trivial for all n?

The motivation: I was thinking to consider a NC analogy for singular homology as follows:

A singular object is a continuous map from Delta^{n}\to X. So We have a natural morphism from C(X)\to C(\Delta^{n}).

With some (or a lot of) abuse of notations and equations, the equation \sum t_{i}=1, which defines the standard simplex \Delta^{n}, could be considered as a commutative picture of universal property of Cuntz algebra O_{n}. This is a motivation to construct a complex as followos: We put C_{n}(A)= Free abelian group generated by Hom(A,, O_{n}). Now we can define a complex

....... C_{n}(A) \to C_{n+1}(A) \to....

using n+1 embeddings O_{n} to O_{n+1}

Does this idea leads to triviality?For what type of C* algebras, this construction is useful?

Let f: \R^n -> \R^n be a function. What would you call the function g: x -> x' f(x) ? Furthermore, if F:\R^n -> \R^n is a linear operator and D \in \R^{n X m} a matrix, how would you refer to the operator G: D -> D' F (D) ? Does this appear in a particular physics or engineering context? Note, I use dash ' to mean transpose.

Can you suggest any material, book or paper on connection of Crossed products of C*-algebras and semigroup C*-algebras?

The same as lacI. I am curious about the when dimer tetramer structure forms in vivo, is it a inducible procedure based on the operator sequence?

In the standard proof showing that hermitian operators have real eigenvalues we exploit the symmetry of the operators to show that they only have real eigenvalues.Does this hold for ALL hermitian operators in general?

By space of measurable functions, I mean L_0(m) where m is a non-atomic sigma-finite measure space.

Suppose X be a linear space. My question is: How can we say that X is not reflexive in any norm. Reflexive means X is linearly isometric to its second dual X''

Give a concrete example of linear operator whose resolvent satisfies a weak parabolic estimate in the region Re z \geq - C(1+ | Im z |)^\ alpha, where 0<\alpha<1.

Thanks

We equip $L(H)$ with the natural Jordan product $a \circ b = \frac12 (a b + ba)$ and its natural involution. We consider the Jordan subalgebra of $L(H)$ generated by $e$ and $e^*$.

We know that the Laplace operator -\Delta is a sectorial operator. So, the operator -\Delta(I-\Delta)^{-1} will be a sectorial operator? And, Can we know the spectral of the -\Delta(I-\Delta)^{-1} if we know the spectral of the operator -\Delta?