The increasing complexity of hardware and software environments in high-performance computing poses big challenges on the development of sustainable and hardware-efficient numerical software. This article addresses these challenges in the context of sparse solvers. Existing solutions typically target sustainability, flexibility, or performance, but rarely all of them. Our new library PHIST provides implementations of solvers for sparse linear systems and eigenvalue problems. It is a productivity platform for performance-aware developers of algorithms and application software with abstractions that do not obscure the view on hardware-software interaction. The PHIST software architecture and the PHIST development process were designed to overcome shortcomings of existing packages. An interface layer for basic sparse linear algebra functionality that can be provided by multiple backends ensures sustainability, and PHIST supports common techniques for improving scalability and performance of algorithms such as blocking and kernel fusion. We showcase these concepts using the PHIST implementation of a block Jacobi-Davidson solver for non-Hermitian and generalized eigenproblems. We study its performance on a multi-core CPU, a GPU, and a large-scale many-core system. Furthermore, we show how an existing implementation of a block Krylov-Schur method in the Trilinos package Anasazi can benefit from the performance engineering techniques used in PHIST.