Sums of invariant subspaces

Stegenga, David A.

doi:10.2140/pjm.1977.70.567

Cited by 3 publications

(5 citation statements)

References 11 publications

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“…Stegenga proved that qH°° + A is norm closed if and only if m (supp #) = 0 or 1. Our results are continuations of Stegenga's works [13] concerning the questions mentioned above. The following is a main theorem proved in Section 3 (actually we shall prove it under more general situations).…”

Section: Sarason's Theorem Let Stf Be An Analytic Subalgebra and ^ Bsupporting

confidence: 88%

“…We shall show that the same assertion is true for qsi 4-A. The proof is the same as the one in [13]. From now on, let put X = M(^), the maximal ideal space of % and put & the closure of #J/ 4-A.…”

Section: Backward Shift Invariant Analytic Subalgebrasmentioning

confidence: 54%

“…It is difficult to answer these problems completely. In this paper, we shall study a special type of algebras qH°° + A, where q is an inner function, which is studied by Stegenga [13]. For an inner function q, we put supp q = {X e 3D; there is a sequence {z n } in D such that z w ->\and^(z w )->0}, Then supp q is a closed subset of 3D.…”

Section: Sarason's Theorem Let Stf Be An Analytic Subalgebra and ^ Bmentioning

confidence: 99%

“…Under the above assumptions (a)-(d) for J< ^and q, we will investigate the algebra qsi + A. In [13], Stegenga proved that qH°° 4-^4 is closed if and only if m (supp q) = 0 or 1. We shall show that the same assertion is true for qsi 4-A.…”

Section: Backward Shift Invariant Analytic Subalgebrasmentioning

confidence: 99%

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Algebras of Bounded Analytic Functions containing the Disk Algebra

Izuchi

1986

Can. j. math.

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Let D be the open unit disk and let ∂D be its boundary. We denote by C the algebra of continuous functions on ∂d, and by L∞ the algebra of essentially bounded measurable functions with respect to the normalized Lebesgue measure m on ∂D. Let H∞ be the algebra of bounded analytic functions on D. Identifying with their boundary functions, we regard H∞ as a closed subalgebra of L∞. Let A = H∞ Pi C, which is called the disk algebra. The algebras A and H∞ have been studied extensively [5, 6, 7]. In these fifteen years, norm closed subalgebras between H∞ and L∞, called Douglas algebras, have received considerable attention in connection with Toeplitz operators [12]. A norm closed subalgebra between A and H∞ is called an analytic subalgebra. In [2], Dawson studied analytic subalgebras and he remarked that there are many different types of analytic subalgebras. One problem is to study which analytic subalgebras are backward shift invariant. Here, a subset E of H∞ is called backward shift invariant if

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Section: Sarason's Theorem Let Stf Be An Analytic Subalgebra and ^ Bsupporting

confidence: 88%

Section: Backward Shift Invariant Analytic Subalgebrasmentioning

confidence: 54%

Section: Sarason's Theorem Let Stf Be An Analytic Subalgebra and ^ Bmentioning

confidence: 99%

Section: Backward Shift Invariant Analytic Subalgebrasmentioning

confidence: 99%

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Algebras of Bounded Analytic Functions containing the Disk Algebra

Izuchi

1986

Can. j. math.

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“…It is known that A(D)/ΘH ∞ is a closed subalgebra of H ∞ /ΘH ∞ if and only if A(D) + ΘH ∞ is norm closed in H ∞ . It is also an interesting fact [8,17] We also characterize the latter property in terms of the canonical factorization Θ = S μ B, where…”

Section: Introductionmentioning

confidence: 99%

Norm controlled inversions and a corona theorem for H∞-quotient algebras

Gorkin

Mortini

Nikolski

2008

Journal of Functional Analysis

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Let Θ be an inner function on the unit disc D. We give a description of those Θ for which the quotient algebra H ∞ /ΘH ∞ has no corona with respect to the visible part of its spectrum, that is for which. It happens that this property is equivalent to the norm controlled inversion property for H ∞ /ΘH ∞ , as well as to a kind of weakened Carleson type embedding theorem. The quotient algebra A(D)/ΘH ∞ is also considered. An interpretation of our main results in terms of model operators is given, too.

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Two algebras of bounded functions

Wolff¹

1982

Duke Math. J.

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We will study two function algebras QC and QA which arise when one looks at function theory on the unit disc in (3 from the point of view of functional analysis. Denote bounded measurable functions on the circle z C[[z by L , boundary values of bounded analytic functions on the disc by H , and continuous functions by C; then Sarason [12] showed that the linear span of H and C is a uniformly closed subalgebra of L. Moreover, H oo + C is the smallest such algebra which properly contains H oo. Together with H oo + C one considers QC, the largest C* algebra contained in H + C, and QA QC q H .I n [14], Sarason showed that QC is the intersection of L with the class VMO of functions whose mean oscillations over arcs tend to zero with the length of the arc.Most results of this paper are analogues for QA of known theorems about the disc algebra A. A reference for these is chapter 6 of [9]. Our main lcmma is the following.THEOREM 1. Suppose f L . There is an outer function q QA such that This should be compared with Fatou's theorem that any closed set of measure zero on the circle is the zero set of a function in the disc algebra. Another formulation of Fatou's theorem would be that theorem remains true with QA and QC replaced by A and C, provided that f is continuous off a closed set of measure zero. We prove theorem in section and in section 2 we use it to prove some other results about QA, by modifying the arguments that lead from Fatou's theorem to the other standard theorems about the disc algebra. We then observe that our results can be combined with those of Stegenga [16] to-answer a question of Sarason [13] about H + C. This is theorem 2. This paper is a revision of my 1979 Berkeley thesis. I am grateful to Donald Sarason, my advisor, for valuable help and encouragement.

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Sums of invariant subspaces

Cited by 3 publications

References 11 publications

Algebras of Bounded Analytic Functions containing the Disk Algebra

Algebras of Bounded Analytic Functions containing the Disk Algebra

Norm controlled inversions and a corona theorem for H∞-quotient algebras

Two algebras of bounded functions

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