TY - JOUR
T1 - Finite Blaschke products of contractions
AU - Gau, Hwa Long
AU - Wu, Pei Yuan
N1 - Funding Information:
∗Corresponding author. E-mail addresses: [email protected] (H.-L. Gau), [email protected] (P.Y. Wu). 1 Research supported by the National Science Council of the Republic of China. 2 Research supported by the National Science Council of the Republic of China under research project NSC 90-2115-M-009-022.
PY - 2003/7/15
Y1 - 2003/7/15
N2 - Let A be a contraction on Hilbert space H and φ a finite Blaschke product. In this paper, we consider the problem when the norm of φ(A) is equal to 1. We show that (1) ∥φ(A)∥=1 if and only if ∥A k∥=1, where k is the number of zeros of φ counting multiplicity, and (2) if H is finite-dimensional and A has no eigenvalue of modulus 1, then the largest integer l for which ∥A l∥=1 is at least m/(n-m), where n=dim H and m=dim ker(I-A*A), and, moreover, l=n-1 if and only if m=n-1.
AB - Let A be a contraction on Hilbert space H and φ a finite Blaschke product. In this paper, we consider the problem when the norm of φ(A) is equal to 1. We show that (1) ∥φ(A)∥=1 if and only if ∥A k∥=1, where k is the number of zeros of φ counting multiplicity, and (2) if H is finite-dimensional and A has no eigenvalue of modulus 1, then the largest integer l for which ∥A l∥=1 is at least m/(n-m), where n=dim H and m=dim ker(I-A*A), and, moreover, l=n-1 if and only if m=n-1.
KW - Blaschke product
KW - Compression of the shift
KW - Contraction
KW - Hankel operator
KW - Toeplitz operator
UR - http://www.scopus.com/inward/record.url?scp=0037716328&partnerID=8YFLogxK
U2 - 10.1016/S0024-3795(02)00697-3
DO - 10.1016/S0024-3795(02)00697-3
M3 - 期刊論文
AN - SCOPUS:0037716328
SN - 0024-3795
VL - 368
SP - 359
EP - 370
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
ER -