Operators constructed by means of basic sequences and nuclear matrices

In this work, we establish an approach to constructing compact operators between arbitrary infinite-dimensional Banach spaces without a Schauder basis. For this purpose, we use a countable number of basic sequences for the sake of verifying the result of Morrell and Retherford. We also use a nuclear operator, represented as an infinite-dimensional matrix defined over the space ℓ₁ of all absolutely summable sequences. Examples of nuclear operators over the space ℓ₁ are given and used to construct operators over general Banach spaces with specific approximation numbers.