Absztrakt: | A norm-sequence is a sequence $\rho=\{\rho_n\}_{n\in\bbfN}$ consisting
of nonnegative real numbers such that $\rho(m+n)\le\rho(m)\rho(n)$
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 {\it 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." \par
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. \par
At the end of the paper, some
interesting examples of norm-sequences are constructed. \par
For
Part II see {\it L. K\'erchy} and {\it V. M\"uller} [Acta Sci.
Math. 65, No. 1-2, 131-138 (1999; Zbl 0932.40003)].}
\RV{Daniel Beltita (Bucure\c{s}ti) |