Toeplitz Operators with Essentially Radial Symbols: For Topelitz operators with radial symbols on the disk, there are important results that characterize boundedness, compactness, and its relation to the Berezin transform. The notion of essentially radial symbol is a natural extension, in the context of multiply-connected domains, of the notion of radial symbol on the disk. In this paper we analyze the relationship between the boundary behavior of the Berezin transform and the compactness of <svg style="vertical-align:-4.74141pt;width:16.4px;" id="M1" height="16.237499" version="1.1" viewBox="0 0 16.4 16.237499" width="16.4" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.2375)"> <g transform="translate(72,-59.01)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑇</tspan> </text> <text transform="matrix(1,0,0,-1,-65.07,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝜙</tspan> </text> </g> </g> </svg> when <svg style="vertical-align:-2.29482pt;width:68.787498px;" id="M2" height="16.637501" version="1.1" viewBox="0 0 68.787498 16.637501" width="68.787498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.6375)"> <g transform="translate(72,-58.69)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝜙</tspan> <tspan style="font-size: 12.50px; " x="11.527766" y="0">∈</tspan> <tspan style="font-size: 12.50px; " x="23.568155" y="0">𝐿</tspan> </text> <text transform="matrix(1,0,0,-1,-39.52,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> </text> <text transform="matrix(1,0,0,-1,-34.65,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">Ω</tspan> <tspan style="font-size: 12.50px; " x="13.465731" y="0">)</tspan> </text> </g> </g> </svg> is essentially radial and <svg style="vertical-align:-0.0pt;width:11.75px;" id="M3" height="10.6875" version="1.1" viewBox="0 0 11.75 10.6875" width="11.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.6875)"> <g transform="translate(72,-63.45)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">Ω</tspan> </text> </g> </g> </svg> is multiply-connected domains.