Criteria of regularity for norm-sequences (Q1304746)

A norm-sequence is a sequence \(\rho=\{\rho_n\}_{n\in\mathbb{N}}\) consisting of nonnegative real numbers such that \(\rho(m+n)\leq\rho(m)\rho(n)\) for all \(m,n\). According to a well-known result due to L. J. Wallen, this happens if and only if \(\rho(n)=\|T^n\|\) for every \(n\), where \(T\) is a bounded linear operator on some Banach space (cf. Solution 92 in \textit{Paul R. Halmos} [A Hilbert space problem book, 2nd ed., rev. and enl., Graduate Texts in Mathematics, 19. New York-Heidelberg-Berlin: Springer-Verlag (1982; Zbl 0496.47001)]). In this case one says that \(\rho\) is the norm-sequence of the operator \(T\). In a previous paper [Acta Sci. Math. 63, No. 3-4, 571-605 (1997; Zbl 0893.47006)], the author proved some invariant subspace theorems holding for operators whose norm-sequences are regular. In the author's words: ``Roughly speaking, this property means the existence of a ``sharp'' upper bound function \(p\) of \(\rho\), which is smooth in a certain sense.'' The main aim of the paper under review is to prove two sufficient conditions for a norm-sequence \(\rho\) to be regular. The first of them is stated by means of the so-called derived sequence of \(\rho\), denoted by \(D\rho\) and defined by \((D\rho)(n)=\rho(n)^{1/n}\). It is well known that \(D\rho\) converges to its infimum. Roughly speaking, Theorem 1 of the paper asserts that, for a norm-sequence \(\rho\) to be regular, ``it is enough to assume that the sequence \(\delta=D\rho\) contains sufficiently long decreasing sections, admitting skips in \(n\) of restricted length.'' The second sufficient regularity condition is contained in Theorem 2 and applies to norm-sequences possessing no long decreasing sections, but arbitrarily long increasing sections. At the end of the paper, some interesting examples of norm-sequences are constructed. For Part II see \textit{L. Kérchy} and \textit{V. Müller} [Acta Sci. Math. 65, No. 1-2, 131-138 (1999; Zbl 0932.40003)].