Vector inequalities for two projections in Hilbert spaces and applications. (English) Zbl 1393.46018

Dragomir, Silvestru Sever

Commun. Math. Anal. 20, No. 2, 8-30 (2017).

Summary: We establish some vector inequalities related to Schwarz and Buzano results [M. L. Buzano, Univ. Politec. Torino, Rend. Sem. Mat. 31(1971–72/1972–73), 405–409 (1974; Zbl 0285.46016)]. We show amongst others that in an inner product space \(H\) we have the inequality \[ \frac{1}{4}\left[ \| x\| \| y\| +| \langle x,y\rangle-2\langle Px,y\rangle-2\langle Qx,y\rangle| \right]\geq | \langle QPx,y\rangle| \] for any vectors \(x,y\) and \(P,Q\) two orthogonal projections on \(H\). If \(PQ=0\) we also have \[ \frac{1}{2}\left[ \| x\| \| y\| +| \langle x,y\rangle| \right]\geq | \langle Px,y\rangle+\langle Qx,y\rangle| \] for any \(x,y\in H\).{ }Applications for norm and numerical radius inequalities of two bounded operators are given as well.

MSC:

46C05	Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
26D15	Inequalities for sums, series and integrals
26D10	Inequalities involving derivatives and differential and integral operators

Keywords:

inner product spaces; Schwarz’s inequality; Buzano’s inequality; projection; operator norm; numerical radius

Citations:

Zbl 0285.46016