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This paper investigates the least-squares projection method for boundedlinear operators, which provides a natural regularization scheme by projectionfor many ill-posed problems. Yet, without additional assumptions, theconvergence of this approximation scheme cannot be guaranteed. We reveal thatthe convergence of least-squares projection method is determined by twoindependent factors -- the kernel approximability and the offset angle. Thekernel approximability is a necessary condition of convergence described withkernel $N(T)$ and its subspaces $N(T){\cap}X_n$, and we give several equivalentcharacterizations for it (Theorem 1). The offset angle of $X_n$ is defined asthe largest canonical angle between space $T^*T(X_n)$ and $T^{\dagger}T(X_n)$(which are subspaces of $N(T)^\bot$), and it geometrically reflects the rate ofconvergence (Theorem 2). The paper also presents new observations for theunconvergence examples of Seidman [10, Example 3.1] and Du [2, Example 2.10]under the notions of kernel approximability and offset angle.

On the two mutually independent factors that determine the convergence of least-squares projection method