Generalized Cauchy–Hankel matrices and their applications to subnormal operators

It is well‐known that for a general operator T on Hilbert space, if T is subnormal, then T m is subnormal for all natural numbers m ≥ 1 . It is also well‐known that if T is hyponormal, then T 2 need not be hyponormal. However, for a unilateral weighted shiftW α ≡ shift ( α 0 , α 1 , ⋯ ) , the hyponormality of W α (detected by the conditionα j ≤ α j + 1for all j ≥ 0 ) does imply the hyponormality of every powerW α m( m ≥ 2 ) . Conversely, we easily see that for 0 < a < b < 1 a weighted shiftW α ≡ shift ( b , a , 1 , 1 , ⋯ )is not hyponormal, therefore not subnormal, but W α m is subnormal for all m ≥ 2 . Hence, it is interesting to note when for some m ≥ 2 , the subnormality of T m implies the subnormality of T . In this article, we construct a non trivial large class of weighted shifts W α such that for some m ≥ 2 , the subnormality of W α m guarantees the subnormality of W α . We also prove that there are weighted shifts with non‐constant tail such that hyponormality of a power or powers does not guarantee hyponormality of the original one. Our results have a partial connection to the following two long‐open problems in Operator Theory: (i) characterize the subnormal operators having a square root; (ii) classify all subnormal operators whose square roots are also subnormal. Our results partially depend on new formulas for the determinant of generalized Cauchy–Hankel matrices and on criteria for their positive semi‐definiteness.