TY - JOUR

T1 - Admissible observation operators for the right-shift semigroup

AU - Partington, Jonathan R.

AU - Weiss, George

PY - 2000

Y1 - 2000

N2 - We give a characterization of infinite-time admissible observation operators for the right-shift semigroup on L2[0, ∞). Our main result is that if A is the generator of this semigroup and C is the observation operator, mapping D(A) into the complex numbers, then C is infinite-time admissible if and only if ∥C(sI-A)-1∥≤M/√Re s for all s in the open right half-plane. We derive this using Fefferman's theorem on bounded mean oscillation and Hankel operators. This result solves a special case of a more general conjecture which says that the same equivalence is true for any strongly continuous semigroup acting on a Hilbert space. For normal semigroups the conjecture is known to be true and then it is equivalent to the Carleson measure theorem. We derive some related results and partial results concerning the case when the signals are not scalar but with values in a Hilbert space.

AB - We give a characterization of infinite-time admissible observation operators for the right-shift semigroup on L2[0, ∞). Our main result is that if A is the generator of this semigroup and C is the observation operator, mapping D(A) into the complex numbers, then C is infinite-time admissible if and only if ∥C(sI-A)-1∥≤M/√Re s for all s in the open right half-plane. We derive this using Fefferman's theorem on bounded mean oscillation and Hankel operators. This result solves a special case of a more general conjecture which says that the same equivalence is true for any strongly continuous semigroup acting on a Hilbert space. For normal semigroups the conjecture is known to be true and then it is equivalent to the Carleson measure theorem. We derive some related results and partial results concerning the case when the signals are not scalar but with values in a Hilbert space.

UR - http://www.scopus.com/inward/record.url?scp=0033672136&partnerID=8YFLogxK

U2 - 10.1007/PL00009866

DO - 10.1007/PL00009866

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AN - SCOPUS:0033672136

SN - 0932-4194

VL - 13

SP - 179

EP - 192

JO - Mathematics of Control, Signals, and Systems

JF - Mathematics of Control, Signals, and Systems

IS - 3

ER -