Projections in operator ranges

Corach, Gustavo; Maestripieri, Alejandra; Stojanoff, Demetrio

doi:10.1090/S0002-9939-05-08007-X

Projections in operator ranges
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by Gustavo Corach, Alejandra Maestripieri and Demetrio Stojanoff PDF

Proc. Amer. Math. Soc. 134 (2006), 765-778 Request permission

Abstract:

If $\mathcal {H}$ is a Hilbert space, $A$ is a positive bounded linear operator on $\mathcal {H}$ and $\mathcal {S}$ is a closed subspace of $\mathcal {H}$, the relative position between $\mathcal {S}$ and $A^{-1}(\mathcal {S}^\perp )$ establishes a notion of compatibility. We show that the compatibility of $(A,\mathcal {S})$ is equivalent to the existence of a convenient orthogonal projection in the operator range $R(A^{1/2})$ with its canonical Hilbertian structure.

References

T. Ando, De Branges spaces and analytic operator functions, Research Institute of Applied Electricity, Hokkaido University, Sapporo, Japan, 1990, 1371-1375.
E. Andruchow, Gustavo Corach, and D. Stojanoff, Geometry of oblique projections, Studia Math. 137 (1999), no. 1, 61–79. MR 1735628
Ju. I. Fedorov, Solution of the $\Delta _{2}$ problem for an equation of hyperbolic type, Differentsial′nye Uravneniya (Ryazan) 12 (1978), 127–135 (Russian). MR 529829
Bruce A. Barnes, The spectral properties of certain linear operators and their extensions, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1371–1375. MR 1664321, DOI 10.1090/S0002-9939-99-05326-5
Adi Ben-Israel and Thomas N. E. Greville, Generalized inverses, 2nd ed., CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 15, Springer-Verlag, New York, 2003. Theory and applications. MR 1987382
Carl de Boor, Convergence of abstract splines, J. Approx. Theory 31 (1981), no. 1, 80–89. MR 619809, DOI 10.1016/0021-9045(81)90032-0
Louis de Branges and James Rovnyak, Square summable power series, Holt, Rinehart and Winston, New York-Toronto, Ont.-London, 1966. MR 0215065
Louis de Branges, Factorization in Kreĭn spaces, J. Funct. Anal. 124 (1994), no. 2, 228–262. MR 1289351, DOI 10.1006/jfan.1994.1107
Gustavo Corach, Alejandra Maestripieri, and Demetrio Stojanoff, Oblique projections and Schur complements, Acta Sci. Math. (Szeged) 67 (2001), no. 1-2, 337–356. MR 1830147
G. Corach, A. Maestripieri, and D. Stojanoff, Generalized Schur complements and oblique projections, Linear Algebra Appl. 341 (2002), 259–272. Special issue dedicated to Professor T. Ando. MR 1873624, DOI 10.1016/S0024-3795(01)00384-6
G. Corach, A. Maestripieri, and D. Stojanoff, Oblique projections and abstract splines, J. Approx. Theory 117 (2002), no. 2, 189–206. MR 1924651, DOI 10.1006/jath.2002.3696
G. Corach, A. Maestripieri and D. Stojanoff, A classification of projectors, Banach Center Publications (to appear).
C. A. Desoer and B. H. Whalen, A note on pseudoinverses, J. Soc. Indust. Appl. Math. 11 (1963), 442–447. MR 156199, DOI 10.1137/0111031
Frank Deutsch, The angle between subspaces of a Hilbert space, Approximation theory, wavelets and applications (Maratea, 1994) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 454, Kluwer Acad. Publ., Dordrecht, 1995, pp. 107–130. MR 1340886
Frank Deutsch, Best approximation in inner product spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 7, Springer-Verlag, New York, 2001. MR 1823556, DOI 10.1007/978-1-4684-9298-9
Jacques Dixmier, Étude sur les variétés et les opérateurs de Julia, avec quelques applications, Bull. Soc. Math. France 77 (1949), 11–101 (French). MR 32937, DOI 10.24033/bsmf.1403
R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415. MR 203464, DOI 10.1090/S0002-9939-1966-0203464-1
P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math. 7 (1971), 254–281. MR 293441, DOI 10.1016/S0001-8708(71)80006-3
T. N. E. Greville, Solutions of the matrix equation $XAX=X$, and relations between oblique and orthogonal projectors, SIAM J. Appl. Math. 26 (1974), 828–832. MR 347845, DOI 10.1137/0126074
Seppo Hassi and Kenneth Nordström, On projections in a space with an indefinite metric, Linear Algebra Appl. 208/209 (1994), 401–417. MR 1287361, DOI 10.1016/0024-3795(94)90452-9
Victor Havin and Burglind Jöricke, The uncertainty principle in harmonic analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 28, Springer-Verlag, Berlin, 1994. MR 1303780, DOI 10.1007/978-3-642-78377-7
Saichi Izumino, Convergence of generalized inverses and spline projectors, J. Approx. Theory 38 (1983), no. 3, 269–278. MR 705545, DOI 10.1016/0021-9045(83)90133-8
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
M. Krein, The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I, Rec. Math. [Mat. Sbornik] N.S. 20(62) (1947), 431–495 (Russian, with English summary). MR 0024574
Zbigniew Pasternak-Winiarski, On the dependence of the orthogonal projector on deformations of the scalar product, Studia Math. 128 (1998), no. 1, 1–17. MR 1489458, DOI 10.4064/sm-128-1-1-17