Paper Title:

Preserver of Local Spectrum of Skew-product Operators

Author(s):

Rohollah Parvinianzadeh^1,*, Meysam Asadipour² and Jumakhan Pazhman³

¹Department of Mathematics,
College of Sciences,
University of Yasouj,
Yasouj, 75918-74934,
Iran.
E-mail: r.parvinian@yu.ac.ir

²Department of Mathematics,
College of Sciences,
University of Yasouj,
Yasouj, 75918-74934,
Iran.
E-mail: Asadipour@yu.ac.ir

³Department of Mathematics,
Ghor Institute of higher education,
Afghanistan.
E-mail: jumapazhman@gmail.com

Abstract:

Let H and K be infinite-dimensional 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 h₀∈ H and k₀∈ K, we show that if two maps φ₁ and φ₂ from B(H) into B(K) satisfy

σ_{φ1(T)φ2(S)*}(k₀)= σ_TS*(h₀})

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 φ₁(T) = PTQ and φ₂(T)^* =Q^-1T^*P^-1 for all T ∈ B(H). Also, we obtain some interesting results in this direction.

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