Characterization of derivations on strongly double triangle subspace lattice algebras by local actions

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">D</m:mi> </m:math> D be a strongly double triangle subspace lattice on a nonzero complex reflexive Banach space and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mtext>Alg</m:mtext> <m:mi mathvariant="script">D</m:mi> </m:math> AlgD the associated subspace lattice algebra. Assume that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>F</m:mi> <m:mo>,</m:mo> <m:mi>G</m:mi> <m:mo>∈</m:mo> <m:mtext>Alg</m:mtext> <m:mi mathvariant="script">D</m:mi> </m:math> F,G∈ AlgD and set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">R</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>F</m:mi> <m:mo>,</m:mo> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:mtext>Alg</m:mtext> <m:mi mathvariant="script">D</m:mi> <m:mo>×</m:mo> <m:mtext>Alg</m:mtext> <m:mi mathvariant="script">D</m:mi> <m:mo>:</m:mo> <m:mi>A</m:mi> <m:mi>F</m:mi> <m:mi>B</m:mi> <m:mo>=</m:mo> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> R(F,G)=\(A,B)∈ AlgD×AlgD:AFB=G\ . A linear mapping <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>δ</m:mi> <m:mo>:</m:mo> <m:mtext>Alg</m:mtext> <m:mi mathvariant="script">D</m:mi> <m:mo>→</m:mo> <m:mtext>Alg</m:mtext> <m:mi mathvariant="script">D</m:mi> </m:math> δ :AlgD→ AlgD is called to be derivable on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">R</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>F</m:mi> <m:mo>,</m:mo> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> R(F,G) if δ ( AB ) = δ ( A ) B + Aδ ( B ) whenever <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:mi mathvariant="script">R</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>F</m:mi> <m:mo>,</m:mo> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (A,B)∈ R(F,G) . In this paper, it is shown that if G is an injective operator or an operator with dense range, then every linear mapping derivable on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">R</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>I</m:mi> <m:mo>,</m:mo> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> R(I,G) <jats:inline-graphic xmlns:xlink="http://www.w3.org/199