Vasile Lauric

Bio: Vasile Lauric is an academic researcher from Florida A&M University. The author has contributed to research in topics: Operator theory & Operator (computer programming). The author has an hindex of 2, co-authored 24 publications receiving 22 citations.

On a weighted toeplitz operator and its commutant

Abstract: We study the structure of a class of weighted Toeplitz operators and obtain a description of the commutant of each operator in this class. We make some progress towards proving that the only operator in the commutant which is not a scalar multiple of the identity operator and which commutes with a nonzero compact operator is zero. The proof of the main statement relies on a conjecture which is left as an open problem.

$(\mathcal {C}_{p}, \alpha )$-hyponormal operators and trace-class self-commutators with trace zero

Abstract: We define the class of (C p ,α)-hyponormal operators and study the inclusion between such classes under various hypotheses for p and α, and then obtain some sufficient conditions for the self-commutator of the Aluthge transform T = |T| 1 2 U|T| 2 of (Cp, α)-hyponormal operators to be in the trace-class and have trace zero.

A remark on λ-interwining hyponormal operators

Abstract: We extend a result concerning λ-commuting normal operators with empty point spectrum. More precisely, we prove that for a hyponormal operator T with empty point spectrum for which there exists a Hilbert-Schmidt operator K such that TK = λKT + μK for some |λ| < 1 and μ ∈ C, implies K = 0.

Remarks on hyponormal operators and almost normal operators.

Abstract: In 1984 M. Putinar proved that hyponormal operators are sub-scalar operators of order two. The proof provided a concrete structure of such operators. We will use this structure to give a sufficient condition for hyponormal operators T with trace-class commutator to admit a direct summand S, so that T\oplus S is the sum of a normal operator and a Hilbert-Schmidt operator. We will investigate what this sufficient condition amounts to in the case of a weighted shift operator.

Some remarks about almost semi-hyponormal operators

Abstract: We define the class of almost semi-hyponormal operators on a Hilbert space and provide some sufficient conditions in which such operators are almost normal, that is their self-commutator is in the trace-class. Mathematics Subject Classification: 47B20

Some concrete operators and their properties

Abstract: We consider integration and double integration operators, the Hardy operator, and multiplication and composition operators on Lebesgue space $L_{p}\left[ 0,1\right] $ and Sobolev spaces $W_{p}^{\left( n\right) }\left[ 0,1\right] $ and $W_{p}^{\left( n\right) }\left( \left[ 0,1\right] \times\left[ 0,1\right] \right) ,$ and we study their properties. In particular, we calculate norm and spectral multiplicity of the Hardy operator and some multiplication operators, investigate its extended eigenvectors, characterize some composition operators in terms of the extended eigenvectors of the Hardy operator, and calculate the numerical radius of the integration operator on the real $L_{2}\left[ 0,1\right] $ space. The main method for our investigation is the so-called Duhamel products method. Some other questions are also discussed and posed.

Properties of the slant weighted Toeplitz operator

Abstract: If = h nin2Z is a sequence of positive numbers, then a slant weighted Toeplitz operator A is an operator on L 2 ( ) defined as A = WM where M is the multiplication operator on L 2 ( ). When the sequence 1, this operator reduces to the ordinary slant Toeplitz operator given by M.C. Ho in 1996. In this paper, we study some algebraic properties of the slant weighted Toeplitz operator. We also obtain its matrix characterization and discuss the adjoint of this operator.

Generalised Slant Weighted Toeplitz Operator

Abstract: A slant weighted Toeplitz operator Aφ is an operator on L(β) defined as Aφ = WMφ where Mφ is the weighted multiplication operator and W is an operator on L(β) given by We2n = βn β2n en, {en}n∈Z being the orthonormal basis. In this paper, we generalise Aφ to the k-th order slant weighted Toeplitz operator Uφ and study its properties. Keywords—Slant weighted Toeplitz operator, weighted multiplication operator.

Monotonicity of generalized Furuta type functions

Abstract: The monotonicity of generalized Furuta type operator function Fs0 (r,s) =C −r 2 (C r 2 (A t 2 BpA t 2 )sC −r 2 ) (p+t)s0+r (p+t)s+r C −r 2 is discussed via the equivalent relations between operator inequalities. Let −1 t 0 . It is shown that, for each s0 such that t p+t < s0 , the function Fs0 (r,s) is decreasing for both r −t and s max{1,s0} . Moreover, some examples are given which imply that, for each s0 1 and r −t , the monotone interval [s0,∞) of s in Fs0 (r,s) is unique in the interval [− r p+t ,∞) . Mathematics subject classification (2010): 47A63, 47B15, 47B65.

On weighted slant Hankel operators

Abstract: Inthispaper, weintroduceandstudythenotionofweightedslantHankeloperatorK β ϕ , ϕ ∈ L ∞ (β) on the space L 2 (β), β = {βn}n∈Z being a sequence of positive numbers with β0 = 1. In addition to some algebraic properties, the commutant and the compactness of these operators are discussed. 1. Preliminaries and Introduction Laurent operators (8) or multiplication operators Mϕ(f 7→ ϕf) on L 2 (T) induced by ϕ ∈ L ∞ (T), T being the unit circle, play a vital role in the theory of operators with their tendency of inducing various classes of operators. In the year 1911, O. Toeplitz (15) introduced the Toeplitz operators given as Tϕ = PMϕ, where P is an orthogonal projection of L 2 (T) onto H 2 (T) and later in 1964, Brown and Halmos (4) studied algebraic properties of these operators. We refer (1, 2, 4, 7, 9 and 12) for the applications and extensions of study to Hankel operators, slant Toeplitz operators, slant Hankel operators and k th -order slant Hankel operators. In the mean time, the notions of weighted sequence spaces H 2 (β) and L 2 (β) also gained momentum. Shield (14) made a systematic study of the Laurent operators on L 2 (β). We prefer to call the Laurent operator on L 2 (β) as weighted Laurent operator. Weighted Toeplitz operators, Slant weighted Toeplitz operators and weighted Hankel operaots on L 2 (β) are discussed in (10), (3) and (5, 6) respectively. In this paper, we extend the study to a new class of operators namely, weighted slant Hankel operators and describe its algebraic properties. We now begin with the notations and preliminaries that are needed in the paper. We consider the space L 2 (β) of all formal Laurent series f(z) = ∑ n∈Z a nz n , an ∈ C, (whether or not the series