On the Limit-Classifications of Even and Odd-Order Formally Symmetric Differential Expressions

Proceedings Mathematical Sciences

V Krishna Kumar³  A Padmanabhan¹  K V Alice² 

[1] Department of Mathematics, Govt. College, Mokeri , India$$;Department of Mathematics, Newman College, Thodupuzha , India$$;Department of Mathematics, University of Calicut , India$$

关键词: Limit classification;    minimal and maximal closed operators;    symmetric operators;    self-adjoint operators;    quotient space $D(T_{max})/D(T_{min})$.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

In this paper we consider the formally symmetric differential expression $M[cdot p]$ of any order (odd or even) ≥ 2. We characterise the dimension of the quotient space $D(T_{max})/D(T_{min})$ associated with $M[cdot p]$ in terms of the behaviour of the determinants$$detlimits_{r,sin N_n}[[f_r g_s](∞)]$$where 1 ≤ 𝑛 ≤ (order of the expression +1); here $[fg](∞) = limlimits_{x→∞}[fg](x)$, where $[fg](x)$ is the sesquilinear form in 𝑓 and 𝑔 associated with 𝑀. These results generalise the well-known theorem that 𝑀 is in the limit-point case at ∞ if and only if $[fg](∞)=0$ for every $f, g in$ the maximal domain 𝛥 associated with 𝑀.

【 授权许可】

Unknown