Relaxing the positivity assumption on the symbol of a Bergman-Toeplitz operator

We examine Toeplitz operators on weighted Bergman spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript script upper V Superscript p"> <mml:semantics> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">A_\mathcal {V}^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 &gt; p &gt; \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , on the unit disk of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with symbols satisfying the simple geometric condition that the values are contained in an angle with vertex in the origin and magnitude less than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The condition is used to relax the conventional positivity assumption of the symbol, yet it is possible to give characterizations of the boundedness and compactness of such Toeplitz operators. The radial weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defining the space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript script upper V Superscript p"> <mml:semantics> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">A_\mathcal {V}^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> may be doubling or exponentially decreasing, but the geometric condition depends only on the symbol and not on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It is known that there are significant differences between doubling and exponential weights for example as regards to the boundedness of Bergman projections. Nevertheless, we give a unified approach which includes both weight classes.