TY - JOUR

T1 - Invariant subspaces of the shift operator. Axiomatic approach

AU - Александров, Алексей Борисович

PY - 1983/8

Y1 - 1983/8

N2 - There is axiomatically described the class of spaces Υ (resp. Χ) of functions, analytic in the unit disk, for which the invariant subspaces of the shift operator f (z) → z f (z) (resp. the inverse shift f(z)→z-1(f(z)-f (0))) are constructed just like the Hardy space H2. It is proved that as Χ one can take, for example, the space H1, the disk-algebra CA, the space UA of all uniformly convergent power series; and as Υ the space of integrals of Cauchy type L1/H-1, the space VMOA. There is also obtained an analog for the space UA of W. Rudin's theorem on z-invariant subspaces of the space CA.

AB - There is axiomatically described the class of spaces Υ (resp. Χ) of functions, analytic in the unit disk, for which the invariant subspaces of the shift operator f (z) → z f (z) (resp. the inverse shift f(z)→z-1(f(z)-f (0))) are constructed just like the Hardy space H2. It is proved that as Χ one can take, for example, the space H1, the disk-algebra CA, the space UA of all uniformly convergent power series; and as Υ the space of integrals of Cauchy type L1/H-1, the space VMOA. There is also obtained an analog for the space UA of W. Rudin's theorem on z-invariant subspaces of the space CA.

UR - http://www.scopus.com/inward/record.url?scp=34250146655&partnerID=8YFLogxK

U2 - 10.1007/BF01882574

DO - 10.1007/BF01882574

M3 - Article

AN - SCOPUS:34250146655

VL - 22

SP - 1695

EP - 1708

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -