### Operators with compatible ranges in an algebra generated by two orthogonal projections

Ilya M Spitkovsky

#### Abstract

The criterion is obtained for operators $A$ from the algebra generated by two orthogonal projections $P,Q$ to have a compatible range, i.e., coincide with the hermitian conjugate of $A$ on the orthogonal complement to the sum of their kernels. In the particular case of $A$ being a polynomial in $P,Q$, some easily verifiable conditions are derived.

#### Article information

Source
Adv. Oper. Theory, Volume 3, Number 1 (2018), 117-122.

Dates
Accepted: 25 April 2017
First available in Project Euclid: 5 December 2017

https://projecteuclid.org/euclid.aot/1512497955

Digital Object Identifier
doi:10.22034/aot.1702-1111

Mathematical Reviews number (MathSciNet)
MR3730342

Zentralblatt MATH identifier
06804319

#### Citation

Spitkovsky, Ilya M. Operators with compatible ranges in an algebra generated by two orthogonal projections. Adv. Oper. Theory 3 (2018), no. 1, 117--122. doi:10.22034/aot.1702-1111. https://projecteuclid.org/euclid.aot/1512497955

#### References

• A. B öttcher and I. M. Spitkovsky, A gentle guide to the basics of two projections theory, Linear Algebra Appl. 432 (2010), no. 6, 1412–1459.
• M. S. Djikić, Operators with compatible ranges, arXiv.math.FA/1609.04884v1.
• R. Giles and H. Kummer, A matrix representation of a pair of projections in a Hilbert space, Canad. Math. Bull. 14 (1971), no. 1, 35–44.
• P. L. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389.
• I. M. Spitkovsky, Once more on algebras generated by two projections, Linear Algebra Appl. 208/209 (1994), 377–395.
• ––––, On polynomials in two projections, Electron. J. Linear Algebra 15 (2006), 154–158.