Errata for Spectral Theory of Operators on Hilbert Spaces

p.5, line-17 (displayed formula). Replace “∑_{i=0}^{∞}” with “∑_{k=0}^{∞}” p.9, lines 2,8. Replace twice “on” with “onto” p.18, line 2. Replace “On” with “In” p.20, line 8. Replace twice “ T^{∗} ” with “ T ” p.23, line 3. Replace “on” with “onto” p.23, line 2. Inset “a” [...of a normed space...] p.24, line 3. Replace “W |_{N(W)^{⊥}}” with “W |_{N (W)^{⊥}}” p.24, line -5. Replace twice. “dim(X) < ∞” with “dim X < ∞” p.25, line 12. Replace “on” with “onto” p.25, line -12. Replace “Theorem 5.73” with “Proposition 5.73” p.33, line -9. Delete “)” [ν ∈ ℂ\σ_{AP} (T)... ] p.35, line 16. Replace “−1^{n+1}” with “(−1)^{n+1}” p.42, line 10. Replace “⟨T x_{n} x_{n}⟩ → λ” with “⟨T x_{n}; x_{n}⟩ → λ” p.43, line -10. Replace “λ = ∥T ∥” with “|λ| = ∥T ∥” p.45, line -4. Replace “bases” with “basis” p.49, lines 12,13 (displayed formulas). Replace twice “k ≥1” with “k ≥ 0” p.49, line -5. Replace “⨁_{k=0}^{∞} H_{k}” with “⨁_{k=−∞}^{∞} H_{k}” p.50, lines -7,-1. Replace twice “stabilities” with “stability” p.51, lines -7,-1. Replace “an” with “a” [a homomorphism] p.51, line -2. Replace “S T − T S” with “S T = T S” p.52, line -15. Insert “)” [6.14])... ] p.52, line 10. (displayed formula). Replace “σ(T)={Φ... ” with “σ′(T)={Φ...” p.58, lines 26,27 (Notes). Replace “[63, Problem 6.14]” with “[66, Problem 6.14]” p.59, lines 6,9,12. Replace three times “sup_{k≥n}” with “sup_{k≥n+1}” p.60, line -13. Replace “a” with “an” [an orthonormal] p.63, line 18. Replace “compact” with “compact normal” p.66, line -10 (displayed formula). Replace “\overline{f(y, x)}” with “\overline{⟨F y; x⟩}” p.67, line 3. Replace “of π with respect to ν” with “of ν with respect to π” p.67, line 12 (displayed formula). Delete “= p(F^{∗}, F)” p.67, line 16. Delete “= p(F^{∗}, F)” p.69, line -7 (displayed formula). Replace ‘‘∥Up∥_{2}^{2}” with “∥p∥_{2}^{2}” p.71, line 15. Inset “)” [in ℂ). Thus ...] p.71, line -13. Insert “be” [— can be viewed as... ] p.71, line -10. Replace “is bounded” with “ ; disjoint union” p.71, line -5. Replace “disjoint countable unions” with “countable disjoint unions” p.71, line -3. Replace “disjoint” with “countable” [any countable] p.72, line 1. Replace “Λ_{j} = Ω_{j} so that” with “if Λ_{j} = Ω_{j}, then” p.72, lines 3 to 6. Replace the paragraph (4 lines) as follows:

Let φ: Ω→ℂ be the function obtained from the identity functions φ_{γ}: Ω_{γ} →Ω_{γ} as follows. If λ lies in the disjoint union Ω = ⋃_{γ∈Γ} Ω_{γ}, then λ ∈ Ω_{β} for some β ∈ Γ, and so set φ(λ) = φ_{β}(λ) = λ for every β ∈ Γ such that λ ∈ Ω_{β}. Since Ω_{β} = σ(T |_{M_{β}}) ⊆ σ(T), it follows that φ is bounded (i.e., φ ∈ L^{∞}(Ω, μ)) with

p.72, line -7. Replace W ψ(λ) = ψ_{β}(λ) with (W ψ)(λ) = ψ_{β}(λ) p.72, line -3. Replace “(∑_{γ∈Γ} φ_{γ}ψ_{γ})_{β}(λ)” with “(W ∑_{γ∈Γ} φ_{γ}ψ_{γ})(λ)” p.73, line 10. Replace “by” with “compare with” [compare with Proposition...] p.73, line 11. Replace ”, and hence Ω is bounded.” with “. Note that φ is bounded.” p.75. Statement of Corollary 3.14, line 4. Replace

“The following assertions are pairwise equivalent.”

with “Consider the following assertions.”

p.75. Statement of Corollary 3.14, line 9. Replace

“Moreover, if any of the above equivalent assertions holds true, then the con-”

with “Claim: (c)⟺(d) ⟹ (a) ⟹ (b). Moreover, if (c) holds true, then the con-”

p.75. Replace proof of Corollary 3.14 as follows:

Proof . By Theorem 3.11, T is unitarily equivalent to a multiplication operator M_{φ} on L^{2}(Ω, μ) for some φ in L^{∞}(Ω, μ). Let U be a unitary transformation in B[L^{2}(Ω, μ), H] such that U^{∗}T U = M_{φ}. Thus ψ in L^{2}(Ω, μ) is a cyclic (starcyclic) vector for the normal operator M_{φ} if and only if x = Uψ in H is a cyclic (star-cyclic) vector for the normal operator T (in fact, ⋁_{n}{T^{n}x} = U ⋁_{n}{M_{φ}^{n}ψ} and ⋁_{m,n}{T^{n}T^{∗m}x} = U ⋁_{m,n}{M_{φ}^{n}M_{φ}^{∗m}ψ}). Corollary 3.13 says that (a) ⟹ (b), and the Remark preceding Corollary 3.13 shows that (c) ⟺ (d) ⟹ (a). Now if (c) holds, then according to part (a) in the proof of Theorem 3.11, μ is finite and φ: Ω → Ω is the identity function (φ(λ) = λ) on the compact set Ω = σ(T). A trivial induction shows that M_{φ}^{n}ψ = φ^{n}ψ and M_{φ}^{∗m}ψ = \overline{φ}^{m}ψ, and so ⋁_{m,n}\{M_{φ}^{n}M_{φ}^{∗m}ψ\} = ⋁_{m,n}\{φ^{n}\overline{φ}^{m}ψ\} = ⋁_{m,n}\{λ^{n}\overline{λ}^{m} ψ\} for every ψ ∈ L^{2}(Ω, μ) and λ in Ω. Since μ is finite, the constant function 1 (1(λ) = 1 for all λ ∈ Ω) lies in L^{2}(Ω, μ) ∩ L^{∞}(Ω, μ). Set ψ = 1 and let P (Ω) denote the set of all polynomials p(·, \overline{·}) : Ω→ ℂ such that λ ↦ p(λ, \overline{λ}). Thus L^{2}(Ω, μ) = P (Ω)^{−} = ⋁_{m,n}\{λ^{n}\overline{λ}^{m} \} = ⋁_{m,n}\{φ^{n}\overline{φ}^{m}1\} = ⋁_{m,n}\{M_{φ}^{n}M_{φ}^{∗m}ψ\} (cf. proof of Theorem 3.11(a)) so that ψ = 1 is a star-cyclic vector for M_{φ}. □

p.76, line 18. Insert “)” [... = C(σ(T)),... ] p.76, line -9. Delete “bounded” p.77, line 13. Insert “]” [... ; (f − ig)⟩] for every... ] p.77, line -8. Replace ‘bounded” with ”included in a disjoint union of σ(T)” p.78, line 9. Replace ‘bounded” with “nonempty” p.78, line 20. Replace “σ(T)” with “Ω” [ in Ω. Note that, for each λ... ] p.80, line 9. Delete “, and since ∪(φ(Λ)^{−}) ⊆ Λ^{−},” p.80, line 10. Replace twice “(Λ^{−})” with “(Λ)” p.80, line 13. Delete “(see [76, Theorem 1.12])” p.84, last line. Insert “the” [... in terms of the orthonormal basis...] p.85, line 5. Insert “in” [A diagonal operator in B[ℓ_{Γ}^{2}]... ] p.85, line 15. Replace “unitary” with “unitarily” [... between the unitarily... ] p.87, line -18. Replace “sequence” with “family” [... bounded family of... ] p.91, line -12. Replace “spaces” with “space” [(of the linear space A onto... ] p.92, item (e). Insert “unital” [... a unital C*-algebra... ] p.94, lines 18,19. Replace “F (Ω) = ℂ^{Ω}” with “F (Ω) ⊆ ℂ^{Ω}”. p.94, line 25. Replace “which is the case of” with

“for instance, if F (Ω) is the algebra of all complex-valued bounded functions on Ω equipped with”

p.95, line 7. Insert “a” [... is not a ∗-algebra ... ] p.95, line -6. Replace “B(σ(T))” with “B(σ(T))” [... ϕ ∈ B(σ(T))... ] p.96, lines 3,4. Replace twice “B(σ(T))” with “B(σ(T))” p.97, line -10. Replace “≤ ∞” with “< ∞” p.98, line 7. Replace “Λ ∈ Ω.” with “Λ ∈ A_{Ω}.” p.102, line 5. Replace “ψ(M_{ϕ})” with “ψ(M_{φ})” [ψ(M_{φ}) = M_{ψ}] p.102, line 13. Replace “Theorem 4.64” with “Theorem 3.64” p.103, line 14. Insert “a” [... with a parameterization... ] p.106, line 5. Replace “then” with “them” [one of them] p.109, line 9. Replace “∫_{Γ} f(λ)” with “∫_{Γ} f(λ) dλ” p.109, line 20 (displayed formula; last symbol). Replace “λ.” with “dλ.” p.109, line 24. Replace “∫_{Γ} f(λ) = 0” with “∫_{Γ} f(λ) dλ = 0” p.109, line -2. Replace “Γ ⊂ Δ ∩ ρ(T) ≠ ∅” with “Γ ⊂ Δ^{−} ∩ ρ(T) ≠ ∅” p.109, line -2. Replace “ (nonempty, since Γ′ and Γ′′ are distinct). Orient...”

with “ If Δ̃ is nonempty, then orient...” p.110, line 1. Replace “ins Γ̃ = Δ̃” with “(ins Γ̃)^{−} = Δ̃^{−}” p.110, lines 5,6. Replace twice “∪” with “∩” p.110, line -3. Replace “in” with “is” p.111, lines 5,6,7,8,9.10,11,12. Replace eight times “(λ − ν)^{−1}” with “(ν − λ)^{−1}” p.112, line -15. Replace “B[H]” with “B[X]” [... :A(σ(T) → B[X]...] p.113, line -5. Replace “A(σ(T)))” with “A(σ(T))” p.113, line -3. Delete “and according to Corollary 2.12(a),” p.114, line 6 (displayed formula). Replace “\frac{1}{2πi} ∥” with “∥\frac{1}{2πi}” p.114, line -2. Replace “A(σ(T))” with “A(σ(T))” p.115, lines 1,4. Replace “A(σ(T))” with “A(σ(T))” p.121, line 8. Replace displayed formula “R_{j}(λ)=R_{T} (λ)E_{Δ_{j}} =R_{T E_{Δ_{j}}} (λ)=R_{T_{j}}(λ)” p.124, line 3. Replace. “dim(N (λI − T)) < ∞” with “dim N (λI − T) < ∞” p.124, line 14.. Replace. “dim(R(E_{Δ_{1}})) < ∞” with “dim R(E_{Δ_{1}}) < ∞” p.124, line -2. Delete “a” [for some positive integer... ] p.124, last line. Replace “N (λI − T)^{n−1}” with “N ((λI − T)^{n−1})” p.125, line 5. Replace “R(E_{Δ})” with “R(E_{λ})” p.125, line 6. Replace “Theorem 4.18” with “Theorem 4.20” p.126, line 2. Insert “a” [on a Hilbert space] p.126, line -3. Replace “B_{δ_{0}}(λ_{0})\{0} =...” with “B_{δ_{0}}(λ_{0})\{λ_{0}} =...” p.127, line 12. Replace “sone” with 1 “some” p.127, line 12. Replace “k < n” with “k < −n” p.127, lines -2. Replace “A(σ(T))” with “A(σ(T))” p.127, last line. Replace “A(σ(T |_{R(E_{Δ})}))” with “A(σ(T |_{R(E_{Δ})}))” p.128, line 2. Replace ”calculus” with ”calculi” p.128, line 3. Replace ”coincides” with ”coincide” p.133, last line. Delete ”.” after □ p.135, line 29. Insert ”)” [... (see, e.g.,[66, Problem 2.17). ] p.137, line -5 (displayed formula). Insert “)” [... ⊖(R(T) + N (S))^{⊥}) ] p.138, lines -12,-11,. Swap in each line “N (S_{+})” with “N (S_{+}^{∗})” p.138, line -10. Swap “F_{ℓ}\F_{r}” with “F_{r}\F_{ℓ}” p.140, lines 6,10. Replace twice ”is” with ”are” [... are such that... ] p.141, line -5. Delete ”)” at the end of the line [... S S^{−1} = I in B[R(S)].] p.143, line 8. Replace ”it” with ”is” [... is a consequence... ] p.144, line -13. Insert ”the” [... in the unital Banach algebra... ] p.147, line 3. Replace ”(λI − T)” with ”λI − T ” p.147, lines 14,16. Replace twice ”B_{∞}[H]/B[H]” with ”B[H]/B_{∞}[H]” p.148, line -7 (displayed formula). Insert “ = σ_{e}(T)\σ_{re}(T),” before “⊆” p.148, line -4 (displayed formula). Insert “ = σ_{e}(T)\σ_{ℓe}(T),” before “⊆” p.149, lines 13,14 (item (e)). Swap “σ_{ℓe}(T)” with “σ_{re}(T)” p.149, line -1, and p.150 lines 1,2. Replace

“. Recall that σ_{ℓe}(T) and σ_{re}(T) are closed in ℂ, and so σ_{+∞}(T) and σ_{−∞}(T) are open in ℂ. Moreover, the expressions for σ_{+∞}(T) and σ_{−∞}(T) and Corollary 5.12 ensure that these sets are subsets of σ_{e}(T).”

with

“, by recalling that σ_{e}(T) = σ_{ℓe}(T) ∪ σ_{re}(T), and also recalling that σ_{e}(T), σ_{ℓe}(T), and σ_{re}(T) are all closed in ℂ, and so σ_{+∞}(T) and σ_{−∞}(T) are open in ℂ.”

p.150, line 10. Replace “0 = dim N (λI −T)” with “0 = dim N (\overline{λ}I −T^{∗})” p.150, line -12. Replace “B_{ε}(λ)\{0}” with “B_{ε}(λ)\{λ}” p.151, line -2. Replace “σ_{P_{4}}” with “σ_{P_{4}}(T)” p.152, line 13,17. Replace twice“σ_{+∞}(T), and σ_{+∞}(T)” with “σ_{+∞}(T), and σ_{−∞}(T)” p.155, line -14. Insert “the” [... in the Calkin algebra... ] p.155, line -7. Replace “dim(R(E_{λ})) < ∞” with “dim R(E_{λ}) < ∞” p.156, line -14. Replace “σ_{0}(T) ∩ π(T) = ∅” with “τ_{0}(T) ∩ π_{0}(T) = ∅” p.156, line -9,10. Replace “Theorem 5.19” with “Corollary 5.18” p.157, line 2. Insert “the” [... a subset of the closed set... ] p.158, line 17. Insert subscript “i” in i=1 [... x = ∑_{i=1}^{n}⟨x; e_{i}⟩e_{i}... ] p.160, lines 17,18. Replace three times “λT − T ” with “λI − T ” p.164, line -7 (Claim 1, line 1). Insert “Suppose R(T^{n}) is closed for every n ∈ ℕ_{0}.” p.165, line 1, Replace “or,” with “(since R(T^{n}) is closed) or,” p.165, line -11. Replace “apply Claim 3 ” with “apply Claims 1 and 3” p.166, lines -12,-11. Replace “(since {R(S^{n})} is nondecreasing).”

with “(S surjective implies dsc(S) = 0).” p.167, line 13. Replace “R(T^{n}) ” with “R(T^{m})” p.169, line -3. Replace “π(T)” with “π_{0}(T)” p.170, line 2. Replace “Corollary 5. 26” with “Corollary 5.26” p.172, line 11. Replace “Also, By” with “ Also, by” p.176, line 25. Replace “Φ ∈ A′′” with “ Φ ∈ Â′′” p.176, line 25. Replace “Φ ∈ A′′\A′′_{F} ” with “ Φ ∈ Â′′\Â′′_{F} ” p.176, line 26. Replace “0 ∉ σ(A^{−1})” with “0 ∉ σ′′(A^{−1})” p.176, line 26. Replace “Φ ∈ A′′_{F} ” with “Φ ∈ Â′′_{F} ” p.176, line 28. Replace “Φ in A′′_{F} ” with “Φ in Â′′_{F} ” p.176, line 30 (displayed formula). Replace twice “Φ ∈ A′′_{F} ” with “Φ ∈ Â′′_{F} ” p.177, line -10. Replace “ensure” with “ensures” p.184, line -8. Replace “B_{ε}(λ)\{0}” with “B_{ε}(λ)\{λ}” p.185, line -9. Delete “W ⊆” [ B ⊆ {T ∈ W: asc(T) < ∞ ⟺ dsc(T) < ∞} ] p.190. Correct reference [57] as follows:

57. T. Ichinose, Spectral properties of tensor products of linear operators I , Trans. Amer. Math. Soc. 235 (1978), 75–113.

p.194, Replace “full spectrum, 52, 87” with “full spectrum, 52, 88”

June 13, 2022