An extension and refinement of the theorems of Douglas and Sebestyén for unbounded operators
An extension and refinement of the theorems of Douglas and Sebestyén for unbounded operators
For a closed densely defined operator $T$ from a Hilbert space $\mathfrak{H}$ to a Hilbert space $\mathfrak{K}$, necessary and sufficient conditions are established for the factorization of $T$ with a bounded nonnegative operator $X$ on $\mathfrak{K}$. This result yields a new extension and a refinement of a well-known theorem of R.G. Douglas, which shows that the operator inequality $A^*A\leq λ^2 B^*B, λ\geq 0$, is equivalent to the factorization $A=CB$ with $\|C\|\leq λ$. The main results give necessary and sufficient conditions for the existence of an intermediate selfadjoint operator $H\geq 0$, such that $A^*A \leq λH \leq λ^2 B^*B$. The key results are proved by first extending a theorem of Z. Sebestyén to the setting of unbounded operators.
Yosra Barkaoui、Seppo Hassi
数学
Yosra Barkaoui,Seppo Hassi.An extension and refinement of the theorems of Douglas and Sebestyén for unbounded operators[EB/OL].(2025-07-17)[2025-08-18].https://arxiv.org/abs/2507.13561.点此复制
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