Subscalarity and invariant subspaces of k-quasi-dominant operators

In [A. Saddi and S. Sbita, Some spectral properties of quasi-dominant operators, Asian-Euro. J. Math., doi:10.1142/S1793557124501304], the authors introduced a new class of operators called, quasi-dominant operators showing that a quasi-dominant operator is not subscalar and asked:” under which condition this operator is subscalar and has Bishop’s property [Formula: see text])?”. In this paper, we continue this study. We introduce new class of operators which we will be called the class of [Formula: see text]-quasi-dominant operators that includes hyponormal operators, [Formula: see text]-hyponormal operators, quasi-dominant and dominant operators. We study various properties of this class of operators. We show that every [Formula: see text]-quasi-dominant operator with [Formula: see text] is subscalar of order 2. As a consequence, we get that if the spectrum of a [Formula: see text]-quasi-dominant operator [Formula: see text] has a nonempty interior in [Formula: see text] and [Formula: see text], then [Formula: see text] has a nontrivial invariant subspace. A part from other results, we show that a [Formula: see text]-quasi-dominant operator [Formula: see text] with [Formula: see text] has Bishop’s property [Formula: see text].