Linear maps preserving idempotency of certain operator products on J$\mathcal{J}$-subspace lattice algebras

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="script">L</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> ${\mathcal{L}}_{1}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="script">L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> ${\mathcal{L}}_{2}$ be <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">J</m:mi> </m:math> $\mathcal{J}$ -subspace lattices on complex Banach spaces X 1 and X 2 , respectively, and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mtext>Alg</m:mtext> <m:msub> <m:mrow> <m:mi mathvariant="script">L</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> $\text{Alg}{\mathcal{L}}_{1}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mtext>Alg</m:mtext> <m:msub> <m:mrow> <m:mi mathvariant="script">L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> $\text{Alg}{\mathcal{L}}_{2}$ be the associated <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">J</m:mi> </m:math> $\mathcal{J}$ -subspace lattice algebras. For an integer k ≥ 2, let ( i 1 , i 2 , …, i m ) be a finite sequence such that { i 1 , i 2 , …, i m } = {1, 2, …, k } and at least one of the terms in ( i 1 , i 2 , …, i m ) appears exactly once. The generalized product of k operators A 1 , A 2 , …, A k is defined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>A</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>A</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>∗</m:mo> <m:mo>…</m:mo> <m:mo>∗</m:mo> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub>