On the geometry of the multiplier space of $\ell_A^p$

For p ∈ (1,∞) \ {2}, some properties of the space Mp of multipliers on l p A are derived. In particular, the failure of the weak parallelogram laws and the Pythagorean inequalities is demonstrated for Mp. It is also shown that the extremal multipliers on the l p A spaces are exactly the monomials, in stark contrast to the p = 2 case.

[1]  R. C. James.  Orthogonality and linear functionals in normed linear spaces , 1947 .

[2]  Remarks on Inner Functions and Optimal Approximants , 2017, Canadian Mathematical Bulletin.

[3]  L. Hörmander,et al.  Estimates for translation invariant operators inLp spaces , 1960 .

[4]  A. Aleman,et al.  Beurling's Theorem for the Bergman space , 1996 .

[5]  Spectra of Inner Functions and l p -Multipliers , 2000 .

[6]  Multipliers of sequence spaces , 2017 .

[7]  J. Mashreghi,et al.  Birkhoff–James Orthogonality and the Zeros of an Analytic Function , 2017 .

[8]  Horst Martini,et al.  On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces , 2012 .

[9]  D. Seco.  A Characterization of Dirichlet-inner Functions , 2017, 1710.02710.

[10]  Raymond Cheng,et al.  Weak parallelogram laws on banach spaces and applications to prediction , 2015, Period. Math. Hung..

[11]  K. Leeuw.  On L p Multipliers , 1965 .

[12]  J. Mashreghi,et al.  Inner Functions in Reproducing Kernel Spaces , 2018, Trends in Mathematics.

[13]  Raymond Cheng,et al.  Duality of the weak parallelogram laws on Banach spaces , 2013 .

[14]  Tadashi Hiraoka.  On Uniformly Convex Spaces , 1953 .

[15]  Mohsen Pourahmadi,et al.  On the geometry of Lp (μ) with applications to infinite variance processes , 2003, Journal of the Australian Mathematical Society.

[16]  W. L. Bynum,et al.  A Weak Parallelogram Law for l p , 1972 .

[17]  H. Shapiro,et al.  Invariant subspaces in Bergman spaces and the biharmonic equation. , 1994 .

[18]  Extremal functions in invariant subspaces of Bergman spaces , 1996 .

[19]  R. Cheng,et al.  On the failure of canonical factorization in ℓAp , 2019, Journal of Mathematical Analysis and Applications.

[20]  A. Olevskiǐ,et al.  Idempotents of Fourier multiplier algebra , 1994 .

[22]  H. Shapiro,et al.  Contractive zero-divisors in Bergman spaces , 1993 .

[23]  S. Richter.  Invariant subspaces of the Dirichlet shift. , 1988 .