Integer Powers of Some Unbounded Linear Operators on P-adic Hilbert Spaces

We initiate and examine integer powers of the (possibly unbounded) diagonal operators on the so-called p-adic Hilbert space Eω (see [10], [11], and [3]). For that, we first give and recall the required background on author’s recent work related to the formalism of unbounded linear operators in the p-adic setting [5]. Next, we shall be dealing with integer powers of the diagonal operators, their product and algebraic sums, and use the definition of integer powers of diagonal operators in order to deal with integer powers to some particular unbounded linear operators on Eω. However, let us mention that our objective in the coming years remains to introduce fractional powers of densely defined closed unbounded linear operators on Eω in order to formulate a p-adic analogue of the classical square root problem of Kato (see [6], [7], [8], and [9]). This is actually, the main motivation of this paper. Let us mention that the p-adic Hilbert space Eω will play a key role throughout the paper. Apart from their intrinsic interests, p-adic Hilbert spaces have found extensive applications in theoretical physics. For more on these and related issues we refer the reader to([10], [11], [3], and [5]) and the references therein. Let K be a complete ultrametric valued field. Classical examples of such a field include Qp the field of p-adic numbers where p ≥ 2 is a prime, Cp the field of p-adic complex numbers, and the field of formal Laurent series([10], [11]). An ultrametric Banach space E over K is said to be a free Banach space (see [10], [11], and [3]) if there exists a family (ei )i∈I (I being an index set) of elements of E such that each element x ∈ E can be written in a unique fashion as