## ABSTRACT

In this chapter, K is invariably a complete, non-trivially valued, non-Archimedean field. The main concern of this chapter is the characterization of the matrix class ( ℓ α , ℓ α ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429281105/d2a97eae-f42e-47ce-9923-c1dd5be90bfe/content/inline-math3_1.jpg"/> , α > 0. When K = R https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429281105/d2a97eae-f42e-47ce-9923-c1dd5be90bfe/content/inline-math3_2.jpg"/> or C https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429281105/d2a97eae-f42e-47ce-9923-c1dd5be90bfe/content/inline-math3_3.jpg"/> , a complete characterization of the matrix class ( ℓ α , ℓ β ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429281105/d2a97eae-f42e-47ce-9923-c1dd5be90bfe/content/inline-math3_4.jpg"/> , α, β ≥ 2, does not seem to be available in the literature. The latest result [4] in this direction characterizes only non-negative matrices in ( ℓ α , ℓ β ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429281105/d2a97eae-f42e-47ce-9923-c1dd5be90bfe/content/inline-math3_5.jpg"/> , when α ≥ β > 1. In case K = R https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429281105/d2a97eae-f42e-47ce-9923-c1dd5be90bfe/content/inline-math3_6.jpg"/> or C https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429281105/d2a97eae-f42e-47ce-9923-c1dd5be90bfe/content/inline-math3_7.jpg"/> , a known simple sufficient condition [7] for a matrix A to belong to ( ℓ α , ℓ α ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429281105/d2a97eae-f42e-47ce-9923-c1dd5be90bfe/content/inline-math3_8.jpg"/> is A ∈ ( ℓ ∞ , ℓ ∞ ) ∩ ( ℓ 1 , ℓ 1 ) . https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429281105/d2a97eae-f42e-47ce-9923-c1dd5be90bfe/content/umath3_1.jpg"/>