The weighted composition-differentiation operator of order n is denoted by D ,,n .In this paper, we investigate some basic properties of compact weighted composition-differentiation operators of order n on the Hardy space.Moreover, we obtain the upper estimate on the norm of the operator D ,,n , in the case that < 1 .

The aim of this paper is to analyze K -frames generated by a bounded linear operator on a separable Hilbert space H . First, we establish some lower bounds for the norm of an operator T when the sequence {T n g} gG,n 0 satisfies the lower K -frame bound for some set G H . Furthermore, we derive a necessary condition for the sequence {T n g} gG,n 0 to be a K -frame.As a consequence, we prove that the hypercyclic operator T with a hypercyclic vector in the range of K cannot generate a K -frame.Additionally, under certain conditions, we construct a Parseval iterative K -frame using an operator.Finally, we determine the form of the K -dual for K -frames generated by an operator.

Let X and Y be two infinite dimensional Banach spaces and B(X ) (resp.B(Y ) ) be the algebra of bounded linear operators on X (resp.Y ).For T B(X ) and x X , T (x) denotes the local spectrum of T at x .Fix an integer k 2 , and let A 1 * A 2 * ... * A k stand for a generalized product of any k operators A 1 , A 2 , ... , A k B(X ) .Given two nonzero vectors x 0 X and y 0 Y , in this paper, we characterize all surjective maps : B(X ) B(Y ) which satisfy A 1 * A 2 * ... * A k (x 0 ) = (A 1 )

In this paper, we study a class of Lie superalgebras based on the Lie algebra sl(2,F) over a field of characteristic not equal to 2. Applying matrix techniques and methods, we determine their automorphisms group and local automorphisms, and characterize their superderivations and local superderivations.

Let A be a unital simple separable infinite dimensional stably finite C*-algebra and B be a large subalgebra of A . In this paper, we show that B has (strong tracial or tracial) mcomparison of positive elements if A has (strong tracial or tracial) m -comparison of positive elements.

The Whittaker Wigner transform (WWT) is a novel addition to the class of Wigner transforms, which has gained a respectable status in the realm of time-frequency signal analysis within a short span of time.Knowing the fact that the study of the time-frequency analysis is both theoretically interesting and practically useful, the aim of this paper is to explore a class of quantitative uncertainty principles associated with the WWT, including the Heisenberg's uncertainty, Benedick's UP, Donoho-Stark's UP, Benedick-Amrein-Berthier UP and the uncertainty principle for orthonormal sequences.

This article includes the core inverses of operators on Hilbert spaces.In addition, the group inverses and Moore-Penrose inverses of these operators are characterised.

The relative distance between eigenvalues of the compression of a not necessarily semibounded self-adjoint operator to a closed subspace and some of the eigenvalues of the original operator in a gap of the essential spectrum is considered. It is shown that this distance depends on the maximal angles between pairs of associated subspaces. This generalises results by Drmač in [Linear Algebra Appl. 244 (1996), 155-163] from matrices to not necessarily (semi)bounded operators.