Some numerical characteristics of direct sum of operators

Authors : Elif OTKUN ÇEVİK

Pages : 1221-1227

Doi:10.31801/cfsuasmas.655136

View : 17 | Download : 11

Publication Date : 2020-12-31

Article Type : Research Paper

Abstract :The numerical range Winsert ignore into journalissuearticles values(T); of a linear bounded operator T on a Hilbert space is the collection of complex numbers of the form insert ignore into journalissuearticles values(Tx,x); with x ranging over the unit vectors in the Hilbert space. In this study, firstly, the connection between numerical range of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the first investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈Linsert ignore into journalissuearticles values(H_{n});, H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈Linsert ignore into journalissuearticles values(H);, then numerical range of the operator A is in the form Winsert ignore into journalissuearticles values(A);= co insert ignore into journalissuearticles values(⋃_{n=1}^{∞}Winsert ignore into journalissuearticles values(A_{n}); );, where co insert ignore into journalissuearticles values(Ω);, Ω⊂C denotes the convex hull of Ω. The numerical radius winsert ignore into journalissuearticles values(T); of a linear bounded operator T on a Hilbert space is a number which is given by winsert ignore into journalissuearticles values(T);=sup {|λ| : λ∈Winsert ignore into journalissuearticles values(T); }. In this study, secondly, the connection between numerical radius of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the second investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈Linsert ignore into journalissuearticles values(H_{n});, H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈Linsert ignore into journalissuearticles values(H);, then numerical radius of the operator A is in the following form winsert ignore into journalissuearticles values(A);= sup winsert ignore into journalissuearticles values(A_{n});. The Crawford number cinsert ignore into journalissuearticles values(T); of a linear bounded operator T on a Hilbert space is given by cinsert ignore into journalissuearticles values(T);=inf {|λ| : λ∈Winsert ignore into journalissuearticles values(T); }. In this study, thirdly, the connection between Crawford number of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the third investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈Linsert ignore into journalissuearticles values(H_{n});, Reinsert ignore into journalissuearticles values(A_{n});≥0 insert ignore into journalissuearticles values(≤0);, H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈Linsert ignore into journalissuearticles values(H);, then Crawford number of the operator A is in the following form cinsert ignore into journalissuearticles values(A);=inf cinsert ignore into journalissuearticles values(A_{n});. Moreover, It is known that for a linear bounded accretive operator T if Winsert ignore into journalissuearticles values(T);⊂{z ∈C:|argz| < ϕ} for any ϕ∈[0, π/ 2);, then it is called a sectorial operator with vertex γ=0 and semi-angle ϕ. In this case, T ∈S_{ϕ}insert ignore into journalissuearticles values(H);. In this study, finally, the connection between sectoriality of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the last investigation the result obtained is as follows: If for any n≥1,H_{n} is a Hilbert space, A_{n}∈Linsert ignore into journalissuearticles values(H_{n});,A_{n}∈S_{ϕ_{n}}insert ignore into journalissuearticles values(H_{n});,H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n},A∈Linsert ignore into journalissuearticles values(H);, then for some ϕ∈[0,π/2);, A∈S_{ϕ}insert ignore into journalissuearticles values(H); the necessary and sufficient condition is sup ϕ_{n}<ϕ. At the end of this study, an example including all these connections has been given. 
Keywords : Direct sum of operators, numerical range, numerical radius, Crawford number, sectorial operator