We show that if $T$ is a bounded linear operator on a complex Hilbert space, then $$\frac{1}{2} ||T|| \leq \sqrt {{\frac{w^2(T)}{2}} + \frac{w(T)}{2} \sqrt{w^2(T) - c^2(T)}} \leq w(T),$$ where $w(\cdot)$ and $c(\cdot)$ are the numerical radius and the Crawford number, respectively. We then apply it to prove that for each $t \in [0, \frac {1}{2})$ and natural number $k$, $$\frac {(1 + 2t)^{\frac{1}{2k}}}{{2}^{\frac{1}{k}}}m(T)\leq w(T),$$ where $m(T)$ denotes the minimum modulus of $T$. Some other related results are also presented.

In this paper, we deal with the problem of characterizing linear maps compressing the numerical range. A counterexample is given to show that such a map need not be a Jordan $*$-homomorphism in general even if the $C^*$-algebras are commutative. Next, under an auxiliary condition we show that such a map is a Jordan $*$-homomorphism.

We present a new approach to sampling theory using the operator theory framework. We use a replacement operator and replace general frames of the sampling and reconstruction subspaces by normalized tight frames. The replacement can be done in a numerically stable and efficient way. The approach enables us to unify the standard consistent reconstruction results with the results for quasiconsistent reconstruction. Our approach naturally generalizes to consistent and quasiconsistent reconstructions from several samples. Not only we can handle sampling problems in a more efficient way, we also answer questions that seem to be open so far.

We continue the analysis of reproducing pairs of weakly measurable functions, which generalize continuous frames. More precisely, we examine the case where the defining measurable functions take their values in a partial inner product space (PIP spaces). Several examples, both discrete and continuous, are presented.

This paper aims to study fixed points in the complete metric space of varieties which are zero at infinity as a subspace of the complete metric space of all varieties. Also, the convex structure of the complete metric space of all varieties will be introduced.

In this note we consider a Durrmeyer type operator having the basis functions in summation and integration due to Mihecşan [Creative Math. Inf. 17 (2008), 466-472.] and Pvǎltvǎnea [Carpathian J. Math. 24 (2008), no. 3, 378-385.] that preserve the linear functions. We present a Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. In the last section of the paper, we obtain the rate of approximation for absolutely continuous functions having a derivative equivalent with a function of bounded variation.

In a recent paper we proposed a possible generalization of L. Schwartz's classical spectral synthesis result for continuous functions in several variables. The idea is based on Gelfand pairs and spherical functions while "translation invariance" is replaced by invariance with respect to the action of affine groups. In this paper we describe the function classes which play the role of the exponential monomials in this setting.