Multipliers between ℓ p spaces

For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo>⩽</m:mo> <m:mi>∞</m:mi> </m:math> 0< p ∞ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>r</m:mi> <m:mo>⩽</m:mo> <m:mi>∞</m:mi> </m:math> 0< r ∞ , 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> {{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> { }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> { }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><</m:mo> <m:mi>r</m:mi> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi>∞</m:mi> </m:math> 0< r< p< ∞ 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> {{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> ({h}₀,{h}₁,{h}₂,… ) 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> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mi>r</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>⁄</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msubsup> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mi>r</m:mi> </m:mrow> </m:msubsup> <m:mo>−</m:mo> <m:msubsup> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>r</m:mi> </m:mrow> </m:msubsup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>⁄</m:mo> <m:mi>r</m:mi> </m:mrow> </m:msup> <m:mo><</m:mo> <m:mi>∞</m:mi> <m:mo>.</m:mo> </m:mrow> </m:math> {({∑ }ₖ₌₀∞{(k+1)}(p-r)/p({h}ₖʳ-{h}ₖ₊₁ʳ))}1/r< ∞ . In that case, the expression on the left is the norm of the multiplier, and it is a compact operator. Further upper and lower bounds are given for the multiplier norm.