TY - JOUR

T1 - Spectrum of J-frame operators

AU - Giribet, Juan

AU - Langer, Matthias

AU - Leben, Leslie

AU - Maestripieri, Alejandra

AU - Martinez Peria, Francisco

AU - Trunk, Carsten

PY - 2018/6/13

Y1 - 2018/6/13

N2 - A J-frame is a frame F for a Krein space (H, [·,·]) which is compatible with the indefinite inner product [·,·] in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in H. With every J-frame the so-called J-frame operator is associated, which is a self-adjoint operator in the Krein space H. The J-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of J-frame operators in a Krein space by a 2×2 block operator representation. The J-frame bounds of F are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the 2×2 block representation. Moreover, this 2×2 block representation is utilized to obtain enclosures for the spectrum of J-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all J-frames associated with a given J-frame operator.

AB - A J-frame is a frame F for a Krein space (H, [·,·]) which is compatible with the indefinite inner product [·,·] in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in H. With every J-frame the so-called J-frame operator is associated, which is a self-adjoint operator in the Krein space H. The J-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of J-frame operators in a Krein space by a 2×2 block operator representation. The J-frame bounds of F are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the 2×2 block representation. Moreover, this 2×2 block representation is utilized to obtain enclosures for the spectrum of J-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all J-frames associated with a given J-frame operator.

KW - frame

KW - Krein space

KW - block operator matrix

KW - spectrum

UR - http://www.opuscula.agh.edu.pl/om-vol38iss5art2

U2 - 10.7494/OpMath.2018.38.5.623

DO - 10.7494/OpMath.2018.38.5.623

M3 - Article

VL - 38

SP - 623

EP - 649

JO - Opuscula Mathematica

JF - Opuscula Mathematica

SN - 1232-9274

IS - 5

ER -