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Cím:Criteria of regularity for norm-sequences.
Szerző:K\'erchy, L\'aszl\'o
Forrás:Integral Equations Oper. Theory 34, No.4, 458-477 (1999).
Nyelv:English
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) 
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