Positivity 12 (2008), 133–150
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010133-18, published online October 29, 2007
Approximation of p-multiplier Operators
via their Spectral Projections
Gerd Mockenhaupt and Werner J. Ricker
Dedicated to the memory of H. H. Schaefer
Abstract. Conditions for a p-multiplier ψ : Z → C are presented which ensure
that the corresponding operator T
(T), can be approximated
by linear combinations of p-multiplier projections coming from the uniform
operator closed, unital algebra of operators generated by T
. Functions of
bounded variation on Z play an important role, as do certain Λ(p)-sets.
Mathematics Subject Classiﬁcation (2000). Primary 42A45; 47A58; 47B40;
Secondary 41A45; 47A25.
Keywords. Fourier p-multiplier, spectral measure, function of bounded
Professor H.H. Schaefer made significant contributions to the area of
“positivity” concerned with spectral measures/Boolean algebras (brieﬂy, B.a.) of
projections. Order properties of the B.a. (with respect to range inclusion) enter
in a crucial way when considering the uniformly (resp. strongly) closed operator
algebra generated by the B.a. of projections. Namely, this operator algebra can
be represented, both topologically and orderwise as a C(K)-space (resp. L
space). These representations become particularly transparent in the presence of a
cyclic vector, as ﬁrst observed by A.I. Veksler, . For the most important contri-
butions of H.H. Schaefer to this area of research we refer to [45, 46, 47], [48, Part
II], , [50, Ch.V, §3], [51, Part IV], , and also to the relevant papers of two of
his former Ph.D. students, namely C. Rall, [34, 35] and B. Walsh, [52, 54, 55, 56].
The last paper by H.H. Schaefer on this topic was together with the second author
of this article, ; see also . H.H. Schaefer was also interested in the spectral
theory and regularity (in the sense of operators on Banach lattices) of certain con-
crete operators arising in harmonic analysis. This began with the Ph.D. thesis of
Former Alexander von Humboldt Fellow at the Universit¨at T¨ubingen, hosted by Prof. H.H.
Schaefer from Sept. 1987 – Feb. 1988.