M. T. Karaev

The numerical radius of a nilpotent operator on a Hilbert space

ABSTRACTL.e t T be a bounded linear operatoro f norm 1 on a Hilbert space H such that T = 0 for some n > 2. Then its numerical radius satisfies w (T) < cos (nn+ 1) and this bound is sharp. Moreover, if there exists a unit vector s E H such that I( TIg) l= cos (n++1 ) then T has a reducings ubspace of dimension n on which T is the usual n-shift. The proofs show that these facts are related to the following result of Fejer: if a trigonometric polynomial f(6) = Zk-!n+i fkeikO is positive, one has If, I < fo cos (n+); moroever, there is essentially one polynomial for which equality holds.

Description of Maximal Ideal Space of Some Banach Algebra with Multiplication as Duhamel Product

Let denote the vector space of complex-valued functions that are continuous on the closed unit disku2009 and have nth order derivatives in D, which can be extended to functions continuous on . Let denote the subspace of the functions which are analytic in D. We prove that is a Banach algebra with multiplication as Duhamel product and describe its maximal ideal space. We also describe commutant and strong cyclic vectors of the integration operator

Functional analysis proofs of Abel's theorems

We give alternative proofs to the classical theorems of Abel, using the concept of Berezin symbol.

On some applications of Duhamel product

Let denote the algebra of all n-times continuously differentiable functions on which are holomorphic on the unit disc : . We prove that is a Banach algebra with multiplication as Duhamel product and describe its maximal ideal space. We also describe the commutant and strong cyclic vectors of the integration operator . Using the Duhamel product we also study the extended eigenvalues and the corresponding extended eigenvectors of the integration operator .

Some results on Berezin symbols

The Berezin symbols are used in the description of Schatten–von Neumann classes σ p , 0<p<∞, in the proof of a unicity theorem of Nikolski and in the approximation problem for inner functions.

SOME RESULTS FOR QUADRATIC ELEMENTS OF A BANACH ALGEBRA

Several properties of some quadratic elements of a unitial Banach algebra are studied. Deddens subspaces are also introduced and discussed.

On Abel Convergence of Double Sequences

In terms of Berezin symbols, we give the concept of (Ber)-convergence of bounded double sequences. We prove that every (Ber)-convergent double sequence is Abel convergent. In particular, by using the Berezin symbols technique, we prove the following double sequence analog of the classical Abel theorem for the sequences: If the sequence regularly converges to L, then

A Characterization of Some Function Classes

We give in terms of Berezin symbols a characterization of Hardy and Besov classes with a variable exponent.

New proof of Nagnibida's theorem

Using the Duhamel product for holomorphic functions we give a new proof of Nagnibida’s theorem on unicellularity of integration operator J α , ( J α f ) ( z ) = ∫ α z f ( t ) d t , acting in the space H o l ( Ω ) .