It is shown that if $α,ζ$ are ordinals such that $1\leq ζ<α<ζω,$ then there is an operator from $C(ω^{ω^α})$ onto itself such that if $Y$ is a subspace of $C(ω^{ω^α})$ which is isomorphic to $C(ω^{ω^α})$ $,$ then the operator is not an isomorphism on $Y.$ This contrasts with a result of J. Bourgain that implies that there are uncountably many ordinals $α$ for which any operator from $C(ω^{ω^α})$ onto itself there is a subspace of $C(ω^{ω^α})$ which is isomorphic to $% C(ω^{ω^α})$ on which the operator is an isomorphism.
In this paper, we introduce a new type of Szasz-Mirakjan operators, which preserve a^x, a > 1 fixed and x\geq 0. We study uniform convergence of the operators by using some auxiliary results and also error estimation is given. The convergence of said operators are shown and analyzed by graphics, also in the same direction, we find a better rate of convergence than Szasz-Mirakjan operators by analyzing the graphics. Voronovskaya-type theorem is studied and a comparison is shown under a sense of convexity with Szasz- Mirakjan operators. In the last section, a modified sequence is constructed in …
In this paper first we describe all (not necessarily linear or bijective) transformations on $\mathbb{R}^d$ with $2\leq d<\infty$ which preserve the area of parallelograms spanned by any two vectors. We also characterize those (not necessarily linear) bijections on an arbitrary real Hilbert space that preserve the latter quantity. This answers a question raised by Rassias and Wagner, and it can be considered as a variant of the famous Wigner theorem on real Hilbert spaces which plays an important role in quantum mechanics. As a consequence, we solve a preserver problem of Molnár and Timmermann…
In this paper general rearrangements of the Haar system in BMO are considered. Several, necessary and suficient, conditions for the boundednes of the induced permutation operator are given. Using analytic families of operators extensions to the case of $L^p$ are obtained .
Let ${\cal H}$ be a Hilbert space, $A$ a positive definite operator in ${\cal H}$ and $\langle f,g\rangle_A=\langle Af,g\rangle$, $f,g\in {\cal H}$, the $A$-inner product. This paper studies the geometry of the set $$ {\cal I}_A^a:=\{\hbox{ adjointable isometries for } \langle \ , \ \rangle_A\}. $$ It is proved that ${\cal I}_A^a$ is a submanifold of the Banach algebra of adjointable operators, and a homogeneous space of the group of invertible operators in ${\cal H}$, which are unitaries for the $A$-inner product. Smooth curves in ${\cal I}_A^a$ with given initial conditions, which are minima…
In this paper we consider the cone of all positive, bounded operators acting on an infinite dimensional, complex Hilbert space, and examine bijective maps that preserve absolute continuity in both directions. It turns out that these maps are exactly those that preserve singularity in both directions. Moreover, in some weak sense, such maps are always induced by bounded, invertible, linear- or conjugate linear operators of the underlying Hilbert space. Our result gives a possible generalization of a recent theorem of Molnar which characterizes maps on the positive cone that preserve the Lebesgu…
Let B(X) be the algebra of all bounded linear operators on a complex Banach space X with dim X greater than 3. In this paper, we characterize the forms of surjective linear maps on B(X) which preserve the dimension of the vector space containing of all fixed points of operators, whenever X is a fifinite dimensional Banach space. Moreover, we characterize the forms of linear maps on B(X) which preserve the vector space containing of all fixed points of operators.
As a continuation of the work on linear maps between operator algebras which preserve certain subsets of operators with finite rank, or corank, here we consider the problem inbetween, that is, we treat the question of preserving operators with infinite rank and infinite corank. Since, as it turns out, in this generality our preservers cannot be written in a nice form what we have got used to when dealing with linear preserver problems, hence we restrict our attention to certain important classes of operators like idempotents, or projections, or partial isometries. We conclude the paper with a …
In this supplement to [GJ1], [GJ3], we give an intrinsic characterization of (bounded, linear) operators on Banach lattices which factor through Banach lattices not containing a copy of $c_0$ which complements the characterization of [GJ1], [GJ3] that an operator admits such a factorization if and only if it can be written as the product of two operators neither of which preserves a copy of $c_0$. The intrinsic characterization is that the restriction of the second adjoint of the operator to the ideal generated by the lattice in its bidual does not preserve a copy of $c_0$. This property of an…
The aim of this paper is to characterize those linear maps from a von Neumann factor $\A$ into itself which preserve the extreme points of the unit ball of $\A$. For example, we show that if $\A$ is infinite, then every such linear preserver can be written as a fixed unitary operator times either a unital *-homomorphism or a unital *-antihomomorphism.
In this paper first we give a partial answer to a question of L. Molnár and W. Timmermann. Namely, we will describe those linear (not necessarily bijective) transformations on the set of self-adjoint matrices which preserve a unitarily invariant norm of the commutator. After that we will characterize those (not necessarily linear or bijective) maps on the set of self-adjoint rank-one projections acting on a two-dimensional complex Hilbert space which leave the latter quantity invariant. Finally, this result will be applied in order to obtain a description of such bijective preservers on the un…
Let $S^1$ be the stopping time space and $\mathcal{B}_1(S^1)$ be the Baire-1 elements of the second dual of $S^1$. To each element $x^{**}$ in the space $\mathcal{B}_1(S^1)$ we associate a positive Borel measure $μ_{x^{**}}$ on the Cantor set. We use the measures $\{μ_{x^{**}}: x^{**}\in\mathcal{B}_1(S^1) \}$ to characterize the operators $T:X\to S^1$, defined on a space $X$ with an unconditional basis, which preserve a copy of $S^1$. In particular, we show that $T$ preserves a copy of $S^1$ if and only if the set $\{μ_{x^{**}}:\;x^{**}\in\mathcal{B}_1(S^1)\}$ is non separable as a subset of $…
We describe the spectrum of weighted $d$-isomorphisms of Banach lattices restricted on closed subspaces that are "rich" enough to preserve some "memory" of the order structure of the original lattice. The examples include (but are not limited to) weighted isometries of Hardy spaces on the polydisk and unit ball in $\mathds{C}^n$.
We prove that Burenkov's Extension Operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n-dimensional Euclidean space.
We prove that Stein's Extension Operator preserves Sobolev-Morrey spaces, that is spaces of functions with weak derivatives in Morrey spaces. The analysis concerns classical and generalized Morrey spaces on bounded and unbounded domains with Lipschitz boundaries in the n-dimensional Euclidean space.
We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of $C_0^\infty(\mathbb{R}^n)$ in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy-Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators.
In this paper, we show a construction of inductive limit for operator system based on Archimedeanization. This inductive limit may be not a closed operator system. We prove that many nuclearity properties could be preserved by a special case of this inductive limit.
An operator $A$ on an $l^p$-space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. The coset of $A$ in the Calkin algebra determines, for example, the Fredholmness of $A$, the Fredholm index, the essential spectrum, the essential norm and the so-called essential pseudospectrum of $A$. This coset can be identified with the collection of all so-called limit operators of $A$. It is known that this identification preserves invertibility (hence spectra). We now show that it also preserves norms and in particular resolvent no…
On a separable C*-algebra A every (completely) bounded map, which preserves closed two sided ideals, can be approximated uniformly by elementary operators if and only if A is a finite direct sum of C*-algebras of continuous sections vanishing at infinity of locally trivial C*-bundles of finite type.
We study infinite order differential operators acting in the spaces of exponential type entire functions. We derive conditions under which such operators preserve the set of Laguerre entire functions which consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane. We obtain integral representations of some particular cases of these operators and apply these results to obtain explicit solutions to some Cauchy problems for diffusion equations with nonconstant drift term.
In a certain sense we generalize the recently introduced and extensively studied notion called quantum Rényi divergence (in another name, sandwiched Rényi relative entropy) and describe the structures of corresponding symmetries. More precisely, we characterize all transformations on the set of density operators which leave our new general quantity invariant and also determine the structure of all bijective transformations on the cone of positive definite operators which preserve the quantum Rényi divergence.
We determine the structure of all bijections on the cone of positive semidefinite operators which preserve the quantum $f$-divergence for an arbitrary strictly convex function $f$ defined on the positive halfline. It turns out that any such transformation is implemented by either a unitary or an antiunitary operator.
