ON SOME COUNTABLY GENERATED ALGEBRA OF ENTIRE FUNCTIONS OF BOUNDED TYPE ON THE SPACE OF ALL BOUNDED SEQUENCES OF COMPLEX NUMBERS AND ITS SPECTRUM
Let _∞ be the complex Banach space of all bounded sequences of complex numbers. In this paper we study the Fr\’echet subalgebra H_bP (_∞) of the Fr\’echet algebra of all entire functions of bounded type H_b(_∞), generated by some fixed countable set P of continuous algebraically independent complex-valued homogeneous polynomials on the space _∞ such that the set P contains an infinite number of elements that share the same degree of homogeneity. We investigate the form of elements of this subalgebra. Furthermore, we show that every linear multiplicative functional, acting from H_bP (_∞) to C, can be uniquely identified with the sequence of its values on the elements of P and we prove that this sequence grows no faster than some geometric progression. We use these results to describe the spectrum of the algebra H_bP (_∞). , number=21(79), journal=PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, author=Vasylyshyn S. І., year=2025, month=Dec., pages=41–51