By Mahowald M., Priddy S. (eds.)

ISBN-10: 0821851020

ISBN-13: 9780821851029

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**Additional resources for Algebraic Topology**

**Example text**

If there exists D ∈ B(H) such that DD∗ = 1 − Td Td∗ and D commutes with T then T has a unitary dilation. Proof. Firstly consider the case when Td is a coisometry. There exists a positive kernel K for T. Fix α in Zd−1 . Deﬁne Kj (α) ∈ B(H), j = 0, 1, . . , by Kj (α) = Tdj K(α)Tdj∗ . Note that Kj (α) ≤ K(α) for all j, and hence b Kj (α) exists. Call this K(α). In this way we deﬁne K : Zd−1 → B(H) that we claim is also a positive kernel for T. To see this, note that then (a) if α is in Zd−1 + K(α) = b Tdj K(α)Tdj∗ = b T α Tdj Tdj∗ = b(T α ) = T α ; and (b) K(α − β)x(β), x(α) α,β∈Zd−1 + K(α − β)Tdj∗ x(β), Tdj∗ x(α) =b ≥0 (since the term in the square brackets is positive for each j).

Gopinath, Determination of vocal-tract shape from impulse response at the lips, J. Acoust. Soc. Am. 49 (1971), 1867–1873. 16 T. M. R. Resnick, The inverse problem for the vocal tract: numerical methods, acoustical experiments, and speech synthesis, J. Acoust. Soc. Am. 73 (1983), 985–1002. N. Stevens, Acoustic phonetics, MIT Press, Cambridge, MA, 1998. W. Symes, On the relation between coeﬃcient and boundary values for solutions of Webster’s Horn equation, SIAM J. Math. Anal. 17 (1986), 1400–1420.

A) There exists a Hilbert space M and a unitary dilation U of T on M ⊕ H ⊕ D such that each U α has the form ⎞ ⎛ ∗ ∗ ∗ ⎝0 T α Dα ⎠ , α ∈ Zd+ . 0 0 ∗ (b) There exists a commuting coisometric extension W of T on H ⊕ D such that each W α has the form Tα 0 Dα , ∗ α ∈ Zd+ . 10) for all α, β ∈ Zd+ . 11) is a positive kernel for T. Proof. (a)⇒(b): Deﬁne W by setting Wj = PH⊕D Uj |H⊕D , j = 1, . . , n. It is straightforward to verify that W has the required properties. (b)⇒(c): Clearly D0 = PH 1|D = 0.

### Algebraic Topology by Mahowald M., Priddy S. (eds.)

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