Besov-Schatten Spaces

Abstract

We introduce the Besov-Schatten spaces <svg style="vertical-align:-4.74141pt;width:42.525002px;" id="M1" height="20.0625" version="1.1" viewBox="0 0 42.525002 20.0625" width="42.525002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.0625)"> <g transform="translate(72,-55.95)"> <text transform="matrix(1,0,0,-1,-71.95,60.73)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐵</tspan> </text> <text transform="matrix(1,0,0,-1,-63.25,57.6)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑝</tspan> </text> <text transform="matrix(1,0,0,-1,-58.47,60.73)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">ℓ</tspan> </text> <text transform="matrix(1,0,0,-1,-47.07,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> </text> <text transform="matrix(1,0,0,-1,-42.19,60.73)"> <tspan style="font-size: 12.50px; " x="0" y="0">)</tspan> </text> </g> </g> </svg>, a matrix version af analytic Besov space, and we compute the dual of this space showing that it coincides with the matricial Bloch space introduced previously in Popa (2007). Finally we compute the space of all Schur multipliers on <svg style="vertical-align:-3.13504pt;width:42.650002px;" id="M2" height="18.0625" version="1.1" viewBox="0 0 42.650002 18.0625" width="42.650002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,18.0625)"> <g transform="translate(72,-57.55)"> <text transform="matrix(1,0,0,-1,-71.95,60.73)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐵</tspan> </text> <text transform="matrix(1,0,0,-1,-63.25,57.6)"> <tspan style="font-size: 8.75px; " x="0" y="0">1</tspan> </text> <text transform="matrix(1,0,0,-1,-58.37,60.73)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">ℓ</tspan> </text> <text transform="matrix(1,0,0,-1,-46.97,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> </text> <text transform="matrix(1,0,0,-1,-42.1,60.73)"> <tspan style="font-size: 12.50px; " x="0" y="0">)</tspan> </text> </g> </g> </svg>.