The angle of an operator and range-kernel complementarity (Q2835257)

Let \(X\) be a complex Banach space with a semi-inner product \([\cdot,\cdot]\) satisfying \([x,x]=\| x\|^2\). Let \(A:X\rightarrow X\) be a bounded linear operator, \(R(A)\) be the range of \(A\) and \(N(A)\) be the kernel of \(A\). In the present paper, the authors prove the following theorem: Theorem 3.4. Let \(A:X\rightarrow X\) be a bounded linear operator with closed range such that \(R(A)+N(A)\) is closed. If NEWLINE\[NEWLINE\Phi (A)=\arccos (\cos A)=\arccos \Big(\inf \Big\{\dfrac{\operatorname{Re}[Ax,x]} {\| Ax\|\cdot \| x\|}:x\notin N(A)\Big\}\Big)<\pi,NEWLINE\]NEWLINE then \(X=R(A)\bigoplus N(A)\). Some applications of this result are given.