


Paper's Title:
Preserver of Local Spectrum of Skewproduct Operators
Author(s):
Rohollah Parvinianzadeh^{1,*}, Meysam Asadipour^{2} and Jumakhan Pazhman^{3}
^{1}Department
of Mathematics,
College of Sciences,
University of Yasouj,
Yasouj, 7591874934,
Iran.
Email: r.parvinian@yu.ac.ir
^{2}Department
of Mathematics,
College of Sciences,
University of Yasouj,
Yasouj, 7591874934,
Iran.
Email: Asadipour@yu.ac.ir
^{3}Department
of Mathematics,
Ghor Institute of higher education,
Afghanistan.
Email: jumapazhman@gmail.com
Abstract:
Let H and K be infinitedimensional complex Hilbert spaces, and B(H) (resp. B(K)) be the algebra of all bounded linear operators on H (resp. on K). For an operator T∈ B(H) and a vector h∈ H, let σ_{T}(h) denote the local spectrum of T at h. For two nonzero vectors h_{0}∈ H and k_{0}∈ K, we show that if two maps φ_{1} and φ_{2} from B(H) into B(K) satisfy
σ_{φ1(T)φ2(S)*}(k_{0})= σ_{TS*}(h_{0}})
for all T, S ∈ B(H), and their range containing all operators of rank at most two, then there exist bijective linear maps P : H→ K and Q : K→ H such that φ_{1}(T) = PTQ and φ_{2}(T)^{*} =Q^{1}T^{*}P^{1} for all T ∈ B(H). Also, we obtain some interesting results in this direction.
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