@article{9bba142cbc894ff0b7e01f20f166e1b7,

title = "Operator ranges and quasicomplemented subspaces of Banach spaces",

abstract = "Given a bounded linear operator T from a separable infinite-dimensional Banach space E into a Banach space Y, an operator range R in E and a closed subspace L ⊂ E such that L ∩ R = {0} and codim (L + R) = ∞, we provide a condition to ensure the existence of an infinite-dimensional closed subspace L1 ⊂ E, containing L as an infinite-codimensional subspace, such that L1 ∩ R = {0} and cl T(L1) = cl T(E). This condition enables us to build closed subspaces of E with a special behaviour with respect to an operator range in E. In particular, we show that if R is an operator range in a Hilbert space, then for every closed subspace H0 in H satisfying H0 ∩ R = {0} and codim(H0 + R) = ∞ there exists an orthogonal decomposition H = V ⊕⊥ W such that V contains H0 as an infinite-codimensional subspace and V ∩ R = W ∩ R = {0}. We also obtain generalizations of some classical results on quasicomplemented subspaces of Banach spaces.",

keywords = "Operator range, Quasicomplemented subspace, Separable Banach space",

author = "Fonf, {V. P.} and S. Lajara and S. Troyanski and C. Zanco",

note = "Funding Information: Research of V. P. Fonf was partially supported by the Israel Science Foundation, Grant # 209/09, and by the Gruppo Nazionale per l'Analisi Matematica, la Probabilit{\`a} e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) of Italy. Research of S. Lajara was partially supported by MICINN projects MTM2014-54182-P, MTM2015-65825-P and MTM2017-86182-P (AEI/FEDER, UE), and by the Fundaci{\'o}n S{\'e}neca (Agencia de Ciencia y Tecnolog{\'i}a de la Regi{\'o}n de Murcia) under project 19275/PI/14. Research of S. Troyanski was partially supported by MICINN projects MTM2014-54182-P and MTM2017-86182-P (AEI/FEDER, UE), by the Fundaci{\'o}n S{\'e}neca (Agencia de Ciencia y Tecnolog{\'i}a de la Regi{\'o}n de Murcia) under project 19275/PI/14, and by the Bulgarian National Scientific Fund, DFNI-I02/10, 2015. Research of C. Zanco was partially supported by a grant of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit{\`a} e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) of Italy, and by the Center for Advanced Studies in Mathematics at the Ben-Gurion University of the Negev, Beer-Sheva, Israel. Publisher Copyright: {\textcopyright} 2019 Instytut Matematyczny PAN.",

year = "2019",

month = jan,

day = "1",

doi = "10.4064/sm180110-31-1",

language = "English",

volume = "246",

pages = "203--216",

journal = "Studia Mathematica",

issn = "0039-3223",

publisher = "Instytut Matematyczny",

number = "2",

}