On the multiplicative behavior of regular matrices

Author:
Robert E. Atalla

Journal:
Proc. Amer. Math. Soc. **26** (1970), 437-446

MSC:
Primary 47.25; Secondary 40.00

DOI:
https://doi.org/10.1090/S0002-9939-1970-0271752-X

Addendum:
Proc. Amer. Math. Soc. **48** (1975), 268.

MathSciNet review:
0271752

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Abstract: Let be a bounded linear operator on compact , with . We define to be the subalgebra of consisting of such that for all , and give a characterization of . We apply the characterization to the multiplicative behavior of regular matrices, considering these as linear operators on . We also relate invariance properties of a matrix under suitable mappings of the integers to topological properties of its support set in , and give an example of a nonnegative multiplicative matrix whose support set is nowhere dense in .

**[1]**R. Atalla and J. Bustoz,*On sequential cores and a theorem of R. R. Phelps*, Proc. Amer. Math. Soc.**21**(1969), 36-42. MR**0243357 (39:4679)****[2]**D. Dean and R. A. Riami,*Permutations with comparable sets of invariant means*, Duke Math. J.**27**(1969), 467-480. MR**0121663 (22:12397)****[3]**P. Erdös and G. Piranian,*The topologization of a sequence space by Toeplitz matrices*, Michigan Math. J.**5**(1958), 139-148. MR**21**#812. MR**0102010 (21:812)****[4]**M. Henriksen,*Multiplicative summability methods and the Stone-Čech compactification*, Math. Z.**71**(1959), 427-435. MR**21**#7434. MR**0108720 (21:7434)****[5]**M. Henriksen and J. R. Isbell,*Multiplicative summability methods and the Stone-Čech compactification*. II, Notices Amer. Math. Soc.**11**(1964), 90-91. Abstract #608-116.**[6]**J. D. Hill and W. T. Sledd,*Approximation in bounded summability fields*, Canad. J. Math.**20**(1968), 410-415. MR**36**#5561. MR**0222510 (36:5561)****[7]**G. M. Petersen,*Summability and bounded sequences*, Proc. Cambridge Philos. Soc.**55**(1959), 257-261. MR**22**#153. MR**0109266 (22:153)****[8]**-,*Regular matrix transformations*, McGraw-Hill, New York and London, 1966. MR**37**#642. MR**0225045 (37:642)****[9]**R. A. Raimi,*Homeomorphisms and invariant measures for*, Duke Math. J.**33**(1966), 1-12. MR**33**#6608. MR**0198450 (33:6608)****[10]**-,*Invariant means and invariant matrix methods of summability*, Duke Math. J.**30**(1963), 81-94. MR**27**#3965. MR**0154005 (27:3965)****[11]**W. Rudin,*Homogeneity problems in the theory of Čech compactifications*, Duke Math. J.**23**(1956), 409-419. MR**18**, 324. MR**0080902 (18:324d)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1970-0271752-X

Keywords:
Regular matrix,
,
multiplicative matrix method,
invariant mean,

Article copyright:
© Copyright 1970
American Mathematical Society