Multipliers between <i>ℓ</i> ^<i>p</i> spaces

Abstract For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>⩽</m:mo> <m:mi>∞</m:mi> </m:math> 0\lt p\leqslant \infty and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>r</m:mi> <m:mo>⩽</m:mo> <m:mi>∞</m:mi> </m:math> 0\lt r\leqslant \infty , the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="fraktur">M</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>r</m:mi> </m:mrow> </m:msub> </m:math> {{\mathfrak{M}}}_{p,r} of (coefficient) multipliers from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:math> {\ell }^{p} to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> <m:mrow> <m:mi>r</m:mi> </m:mrow> </m:msup> </m:math> {\ell }^{r} is completely characterized. This is elementary in most instances. The interesting case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>r</m:mi> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>&lt;</m:mo> <m:mi>∞</m:mi> </m:math> 0\lt r\lt p\lt \infty requires more effort, and it is shown that a sequence of complex numbers belongs to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="fraktur">M</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>r</m:mi> </m:mrow> </m:msub> </m:math> {{\mathfrak{M}}}_{p,r} if and only if the sequence of their absolute values has a nonincreasing rearrangement <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left({h}_{0},{h}_{1},{h}_{2},\ldots ) satisfying <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:msup> <m:mrow> <m:mfenced open="(" close=")"> <m:mrow> <m:munderover> <m:mrow> <m:mo>∑</m:mo> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:munderover> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow>