Closed Range Spectrums Of 3×3 Upper Triangular Operator Matrices With Unbounded Entries

This work mainly deals with the completion problems of closed range spectrums of 3×3 upper triangular operator matrices with unbounded entries in infinite dimensional complex separable Hilbert spaces.Firstly,the possible closed range spectrums of 3 x 3 upper triangular operator matrices MD,E,F with unbounded entries are studied.For given closed operators A,B and C,the union of closed range spectrums of MD,E,F are described.Secondly,the inherent closed range spectrums of 3×3 upper triangular operator matrices MD,E,F with unbounded entries are obtained.For given closed operators A,B and C,the intersection of closed range spectrums of MD,E,F are described.Finally,some examples are presented to verify the effectiveness of the results.