recently, mohiuddine and alghamdi introduced the notion of lacunary statistical convergence in a locally solid riesz space and established some results related to this concept. in this paper, some inclusion relations between the sets of statistically convergent and lacunary statistically convergent sequences are established and extensions of a decomposition theorem, a tauberian theorem to the l...

In this paper we deal with the connection of frames with the class of Hilbert Schmidt operators. First we give an easy criteria for operators being in this class using frames. It is the equivalent to the criteria using orthonormal bases. Then we construct Bessel sequences frames and Riesz bases for the class of Hilbert Schmidt operators using the tensor product of such sequences in the original...

We show that if an operator T is bounded on weighted Lebesgue space L(w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator [b, T ] with a function b in BMO will obey a quadratic bound with respect to the A2 constant of the weight. We also prove that the kth-order commutator T k b = [b, T k−1 b ] will obey a bound that is a power (k + 1) of the A2 consta...

A kind of planar network of strings with non-collocated terms in boundary feedback controls is considered. Suppose that the network is constituted by n non-uniform strings, connected by one vibrating point mass. The displacements of these strings are continuous at the common vertex. The noncollocated terms are contained in feedback controls at exterior nodes. The well-posedness of the correspon...

We study the theory of spectral measures in topological vector spaces. We extend the Hilbert space theory to this setting and generalize the notion of spectral measures in some useful ways to provide a framework for operator theory in this setting. The Riesz representation theorem is proved without assuming local convexity. This theorem is applied to give sufficient conditions for an operator (...