ACTA SCIENTIARUM MATHEMATICARUM最新文献

Let (T=left[ begin{array}{cc} T_{11} &{} T_{12} T_{21} &{} T_{22} end{array} right] ) be accretive-dissipative, where (T_{11},T_{12},T_{21},) and (T_{22} ) are (ntimes n) complex matrices. Let f be a non-negative function on ( [0,infty )) such that (f(0)=0), and let (alpha ,beta in (0,1)) such that (alpha +beta =1). For every unitarily invariant norm (left| left| left| cdot right| right| right| ), it is shown that

whenever (trightarrow fleft( sqrt{t}right) ) is convex and

whenever f is concave.

Let P(z) be a polynomial of degree n, then it is known that for (alpha in {mathbb {C}}) with (|alpha |le frac{n}{2},)

This inequality includes Bernstein’s inequality, concerning the estimate for (|P^prime (z)|) over (|z|le 1,) as a special case. In this paper, we extend this inequality to (L_p) norm which among other things shows that the condition on (alpha ) can be relaxed. We also prove similar inequalities for polynomials with restricted zeros.

We give a characterization of all the unitarily invariant norms on a finite von Neumann algebra acting on a separable Hilbert space. The characterization is analogous to von Neumann’s characterization for the (ntimes n) complex matrices and the characterization in Fang et al. (J Funct Anal 255(1):142–183, 2008) for (II_{1}) factors.

We discuss the question of the analyticity of a rank one perturbation of an analytic operator. If ({mathscr {M}}_z) is the bounded operator of multiplication by z on a functional Hilbert space ({mathscr {H}}_kappa ) and (f in {mathscr {H}}) with (f(0)=0,) then ({mathscr {M}}_z + f otimes 1) is always analytic. If (f(0) ne 0,) then the analyticity of ({mathscr {M}}_z + f otimes 1) is characterized in terms of the membership to ({mathscr {H}}_kappa ) of the formal power series obtained by multiplying f(z) by (frac{1}{f(0)-z}.) As an application, we discuss the problem of the invariance of the left spectrum under rank one perturbation. In particular, we show that the left spectrum (sigma _l(T + f otimes g)) of the rank one perturbation (T + f otimes g,) (,g in ker (T^*),) of a cyclic analytic left invertible bounded linear operator T coincides with the left spectrum of T except the point (langle {f},,{g} rangle .) In general, the point (langle {f},,{g} rangle ) may or may not belong to (sigma _l(T + f otimes g).) However, if it belongs to (sigma _l(T + f otimes g) backslash {0},) then it is a simple eigenvalue of (T + f otimes g).

We prove that the invariant subspaces of the Hardy operator on (L^2[0,1]) are the spaces that are limits of sequences of finite dimensional spaces spanned by monomial functions.

Let B be a proper open subset in ({{mathbb {R}}}^N) and C be a regular cone in ({{mathbb {R}}}^N). On our previous paper of Acta Scientiarum Mathematicarum 85, 595–611 (2019), we have defined the space of generalized Hardy functions, (G_{omega ^*,A}^p(T^B)), (1< p le 2,) and (A ge 0), and have shown that the functions in (G_{omega ^*,A}^p(T^B)) have distributional boundary values in the weak topology of Beurling tempered distributions, ({mathcal {S}}_{(omega )}^prime ). In this paper we show that if the distributional boundary values are convolutors in Beurling ultradistributions of (L_2)-growth, then the functions in (G_{omega ^*,0}^p(T^C)), (1< p le 2,) can be represented as Cauchy and Poisson integral of the boundary values in ({mathcal {S}}_{(omega )}^prime ).

This paper is a continuation of previous works Lahmar (Filomat 36:2551-2572, 2022), Lahmar (Filomat 36: 4575–4590, 2022), Lahmar (Preprint) where we defined a new class of operators called pseudo-generalized invertible operators that includes both the set of generalized invertible operators and the set of Drazin invertible operators. Here we focus essentially on the perturbation problem of pseudo-generalized invertible operators and the particular case of DPG invertibility.

For a closed linear relation everywhere defined on a Hilbert space the concepts of isometry, co-isometry, partial isometry, and generalized inverse are introduced and studied. Part of the results proved in this paper improve and generalize some results known for these concepts. In particular, we extend those of [Acta Sci. Math. (Szeged), 70 (2004), 767–781] and [Studia Math. 205 (2011), no. 1, 71–82].

A best approximation theorem for almost cyclic contractions has been proved in the recent article (Sadiq Basha in J. Fixed Point Theory Appl 23:32, 2021). The purpose of this note is to show that, with the same hypotheses as in the preceding best approximation theorem, the conclusion of the theorem can be strengthened to produce a best proximity point rather than a best approximation and hence a best proximity point theorem for almost cyclic contractions in the framework of a uniformly convex Banach space. Further, it is interesting to observe that such a best proximity point theorem for almost cyclic contractions generalizes/subsumes the well known best proximity point theorem, due to Eldred and Veeramani (J Math Anal Appl 323:1001–1006, 2006), for cyclic contractions in the framework of a uniformly convex Banach space. On the other hand, these best approximation theorems and best proximity point theorems for some types of contractions do not generalize the most elegant Banach’s contraction principle because of the underlying richer framework of a uniformly convex Banach space rather than a simpler framework like a complete metric space. Therefore, the purpose of this note is to bring forth the framework of utmost complete space and establish a best proximity point theorem for almost cyclic contractions in such a simpler framework, thereby generalizing the contraction principle.

We use the q-Duhamel product to provide a Banach algebra structure to some closed subspaces of the Wiener disk- algebra (W_{+}left( mathbb {D}right) ) of analytic functions on the unit disk (mathbb {D}) of the complex plane (mathbb {C.}) We study the q-integration operator on (W_{+}left( mathbb {D}right) ,) namely, we characterize invariant subspaces of this operator and describe its extended eigenvalues and extended eigenvectors. Moreover, we prove an addition formula for the spectral multiplicity of the direct sum of q-integration operator on (W_{+}left( mathbb {D}right) ) and some Banach space operator.