The Product of Automorphic Weighted Composition Operators on Hardy Space H
<sup>2</sup>
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Abstract<p>Let <inline-formula>
<tex-math><?CDATA $n\in {\mathbb{N}},{p}_{i}\in {\rm{U}},{\alpha }_{{P}_{i}}(z)=\frac{{p}_{i}-z}{1-{\bar{p}}_{i}z}(z\in {\rm{U}})$?></tex-math>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mrow>
<mi>n</mi>
<mo>∈</mo>
<mi>ℕ</mi>
<mo>,</mo>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
<mo>∈</mo>
<mi mathvariant="normal">U</mi>
<mo>,</mo>
<msub>
<mi>α</mi>
<mrow>
<msub>
<mi>P</mi>
<mi>i</mi>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
<mo>−</mo>
<mi>z</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<msub>
<mover accent="true">
<mi>p</mi>
<mo>¯</mo>
</mover>
<mi>i</mi>
</msub>
<mi>z</mi>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>∈</mo>
<mi mathvariant="normal">U</mi>
<mo stretchy="false">)</mo>
</mrow>
</math>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn1.gif" xlink:type="simple"></inline-graphic>
</inline-formula>, and let <inline-formula>
<tex-math><?CDATA ${f}_{1}\in {H}^{\infty },i=1,\ldots,n$?></tex-math>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mrow>
<msub>
<mi>f</mi>
<mn>1</mn>
</msub>
<mo>∈</mo>
<msup>
<mi>H</mi>
<mi>∞</mi>
</msup>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…</mo>
<mo>,</mo>
<mi>n</mi>
</mrow>
</math>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn2.gif" xlink:type="simple"></inline-graphic>
</inline-formula>. We discuss the relation between the points <italic>p<sub>i</sub>
</italic> in U and the functions <italic>f<sub>i</sub>
</italic> in U and the properties of the product of automorphic weighted composition operators <inline-formula>
<tex-math><?CDATA ${W}_{{f}_{1},{\alpha }_{{p}_{1}}}\,{W}_{{f}_{2},{\alpha }_{{p}_{2}}}\ldots {W}_{{f}_{i},{\alpha }_{pi}}$?></tex-math>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mrow>
<msub>
<mi>W</mi>
<mrow>
<msub>
<mi>f</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>α</mi>
<mrow>
<msub>
<mi>p</mi>
<mn>1</mn>
</msub>
</mrow>
</msub>
</mrow>
</msub>
<mspace width="0.25em"></mspace>
<msub>
<mi>W</mi>
<mrow>
<msub>
<mi>f</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<msub>
<mi>α</mi>
<mrow>
<msub>
<mi>p</mi>
<mn>2</mn>
</msub>
</mrow>
</msub>
</mrow>
</msub>
<mo>…</mo>
<msub>
<mi>W</mi>
<mrow>
<msub>
<mi>f</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
<msub>
<mi>α</mi>
<mrow>
<mi>p</mi>
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
</math>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn3.gif" xlink:type="simple"></inline-graphic>
</inline-formula> on Hardy space H<sup>2</sup>. In fact, it is very nice connection between analytic function theory and operator theory. In this paper, we give the sufficient and necessary conditions to be normal, unitary, hermitian operator on <italic>H</italic>
<sup>2</sup> and we shall present the shape of the numerical range of it.</p>
