Skew-product and peripheral local spectrum preservers

Consider two infinite-dimentional complex Hilbert spaces, denoted as H and K. Choose two nonzero vectors h0 is an element of H and k0 is an element of K. Let L (H) and L (K) represent the algebra of all bounded linear operators on H and K, respectively. Additionally, gamma T(x) stands for the peripheral local spectrum of an operator T at x, and Fn(K) represents the ideal of operators in L (K) with a rank at most n. Our goal is to demonstrate that if the maps cb1 : L(H) -> L(K) and cb2 : L(H) -> L(K) satisfy the condition gamma TS & lowast; (h0) = gamma phi 1(T)phi 2(S)& lowast;(k0) for all T, S is an element of L (H) and their ranges contain F2(K), then there exist bijective linear operators U : H --> K and V : K --> H such that cb1(T) = UTV and cb2(T) = V -1TU-1 for all T is an element of L(H). Moreover, we derive some interesting results in this direction.