Errata

Preface

▷ p.ix – line 12. Replace  many on occasions  with  on many occasions

Chapter 1

▷ p.2 – line 6 from bottom. Replace hold with holds ▷ p.8 – line 8. Delete “a” [ Let ≈ be an equivalence... ] ▷ p.9 – line 4 from bottom. Replace has lower with has a lower ▷ p.10 – line 5 (above Section 1.6). Insert after “Note that”

if for every pair x_{1}, x_{2} in X either x_{1} ≤ x_{2} or x_{2} ≤ x_{1}, then ▷ p.12 – line 3 (Proof of Theorem 1.2). Replace x ∈ A_{0}A with x ∈ A_{0} ▷ p.12 – line 23 (above Section 1.7). Insert (equipped with its natural ordering)

[ ... integers (equipped with its natural ordering) is well-ordered. ] ▷ p.18 – lines 18, 19 (between Proposition 1.8 and Theorem 1.9). Replace twice

nonempty set with set with more than one element ▷ p.18 – line 23 (between Proposition 1.8 and Theorem 1.9). Replace

nonempty sets with sets with more than one element ▷ p.19 – line 16. Replace , where ∅ ≠ A_{0} ⊆ X with so that A_{0} ⊆ X ▷ p.19 – line 19 (Proof of Theorem 1.9). Replace

which implies that the nonempty set A_{0} is an infinite set. with where A_{0} is an infinite (at least countably infinite) set.

▷ p.24 – line 6 from bottom (Problem 1.2(a)). Replace = with ⊇ ▷ p.26 – line 18 (Problem 1.12). Insert after “proper subset of Y.”

Apply this to show that a finite set has no proper equivalent subset. ▷ p.26 – lines 1–4 from bottom. Displayed formulas in Problem 1.13 should read:

A R_{1} B if A▽B = ∅ (i.e., if A = B),

A R_{2} B if A▽B is finite,

A R_{3} B if A and B are singletons,

A R_{4} B if A ⊆ B or B ⊆ A (i.e., if A\B = ∅ or B\A = ∅), A R_{5} B if A ⊆ B (i.e., if A\B = ∅).

▷ p.28 – line 15 (Problem 1.17(b)). Replace F ∧ G on F with F ∧ G in F ▷ p.32 – lines 4, 5 (Problem 1.26). Replace

nonempty sets with sets with more than one element ▷ p.32 – lines 7, 8 (Problem 1.26(a)). Replace

(nonempty) finite sets  with  finite sets with more than one element

Chapter 2

▷ p.54 – line 10 from bottom (Example 2K). Replace A\{s} with A\{x_{s}} ▷ p.61 – line 17 (Example 2K, 5th displayed equation). Replace y_{j} with y_{i} ▷ p.64 – line 11. Replace denotes with denote ▷ p.65 – line 8. Replace such the diagram with such that the diagram ▷ p.67 – line 6 from bottom. Replace M and M with M and N ▷ p.79 – line 18 (Problem 2.19, first line). Replace let with Let

Chapter 3

▷ p.88 – line 9 (Proposition 3.2). Replace Let S be set with Let S be a set ▷ p.90 – line 4 from bottom (Example 3D). Replace d_{∞} with d_{∞}(x, y) ▷ p.93 – line 4 from bottom. Replace equip with equips ▷ p.93 – line 3 from bottom. Replace rule with role ▷ p.94 – last line (Example 3F). Replace converges with converge ▷ p.96 – lines 1 to 6 (Example 3G). Replace these 6 lines with the next 2 lines.

for every p ≥1. Indeed, take any ε ∈ (0, \frac{1}{2}] and note that g_{n}(\frac{1}{n} + ε) → 0 and ∫_{0}^{1}|g_{n}(s)|^{p} ds ≤ \frac{1}{n} + ε + g_{n}(\frac{1}{n} + ε)^{p}. Hence ∫_{0}^{1}|g_{n}(s)|^{p} ds → 0, and therefore

▷ p.101 – line 6 (below Definition 3.9). Replace x_{0} with u ▷ p.101 – line 7 (below Definition 3.9). Replace exist with exists ▷ p.117 – line 11 from bottom. Replace be open ball with be an open ball ▷ p.122 – line 5 from bottom. Replace (a′),(a),(b)and(c) with (a′),(b),(c)and(d) ▷ p.123 – line 12 from bottom (Example 3P). Replace 3.44(c) with 3.44(b) ▷ p.124 – line 14. Insert if the subcollection is nonempty

[ ... a member of any base for X if the subcollection is nonempty. ] ▷ p.124 – line 11 from bottom (Proof of Proposition 3.34). Replace X with x ▷ p.125 – Proposition 3.37 items (b), (c), (d). Replace three times ∪ with ∩ ▷ p.128 – line 15. Replace in it with in them ▷ p.129 – line 6 (Example 3R). Insert space after “metric” ▷ p.129 – line 6 (Example 3R). Insert . after “complete” ▷ p.132 – line 14 (first displayed formula in Example 3T). Replace d_{X} with d_{Y} ▷ p.135 – line 6 from bottom (Theorem 3.45). Replace 3.14(a) with 3.14(b) ▷ p.138 – line 3 (Corollary 3.47). Delete . after “X” ▷ p.141 – line 14 (Theorem 3.50). Delete of after “F ” and before “over” ▷ p.142 – line 6 (displayed formula). Replace F J̃ with F JJ̃ ▷ p.142 – line 9. Delete a after “another” and before “continuous” ▷ p.143 – last line. Replace meagre with meager ▷ p.144 – line 2. Replace comeagre with comeager ▷ p.144 – line 3. Replace nonmeagre with nonmeager ▷ p.145 – line 4 from bottom (Theorem 3.57). Insert complete before “metric” ▷ p.146 – line 2 (Proof of Corollary 3.57). Replace A\X with X\A ▷ p.146 – line 10. Replace second category in x with second category in X ▷ p.147 – line 19 (Example 3W(a), line 2). Replace 3.56 with 3.57 ▷ p.148 – line 14 (above Section 3.10). Replace Dually, A with Dually, a ▷ p.148 – line 19 (Section 3.10, line 3). Replace of A with of A ▷ p.153 – line 10 (Example 3X). Replace ℵ_{0} ≤ A_{ε} with ℵ_{0} ≤ #A_{ε} ▷ p.156 – line 2 from bottom. Replace (Theorem 3.20) with (Theorem 3.30) ▷ p.157 – line 5 from bottom. Replace then B′ ∈ ... with then B′ ∈ ... ▷ p.160 – line 9 from bottom (Example 3Z(i)). Delete a after “is” ▷ p.161 – line 10 from bottom. Replace according o with according to ▷ p.162 – line 20. Replace d_{Y}((f_{n}^{(n)}(a), f_{n}^{(n)}(x))<ε with d_{Y}(f_{n}^{(n)}(a), f_{n}^{(n)}(x))<ε ▷ p.165 – line 14 (Problem 3.4). Replace p + 1 with p − 1 ▷ p.167 – line 6 (Problem 3.8(b)). Replace R^{p}(S)⊂R^{q}(S) with R^{p}(S)⊂R^{r}(S) ▷ p.167 – line 7 (Problem 3.8(b)). Replace diam(S)^{\frac{p−r}{r}} with diam(S)^{\frac{p−r}{pr}} ▷ p.167 – line 13 (displayed formula). Replace d_{∞} with d_{∞}(x, y) ▷ p.167 – line 13 (displayed formula in Problem 3.9). Replace d_{k} with d_{i} ▷ p.171 – line 2 from bottom. Replace → (C[0, 1], d) with → (C[0, 1], d_{∞}) ▷ p.172 – line 7 from bottom. Replace (C[0, 1]d_{∞}) with (C[0, 1], d_{∞}) ▷ p.172 – lines 4, 6 from bottom. Replace twice (C[0, 1]d_{p}) with (C[0, 1], d_{p}) ▷ p.176 – line 16 (Problem 3.30). Insert continuous before “mapping” ▷ p.181 – lines 7, 6 from bottom. Items (b), (c) in Problem 3.42 should read:

(b) B_{ρ}(x) ⊆ B_{ρ}[x]^{∘} and ∂B_{ρ}[x] ⊆ {y ∈ X: d(y, x) = ρ}.

(c) B_{ρ}(x)^{−} ⊆ B_{ρ}[x] and ∂B_{ρ}(x) ⊆ {y ∈ X: d(y, x) = ρ}.

▷ p.181 – line 2 from bottom (Problem 3.43(a)). Replace second = with ≥ ▷ p.183 – line 18 (Problem 3.48(a)). Replace #A ≤ #B with #B ≤ #A ▷ p.183 – line 18 (Problem 3.48(a)). Replace Problems with Problem ▷ p.184 – line 8 from bottom (Problem 3.50). Insert space after “topological” ▷ p.188 – line 5 (Hint to Problem 3.56(c)). Replace {υ_{n}(k)}_{i∈ℕ} with {υ_{n}(k)}_{k∈ℕ} ▷ p.192 – lines 7, 8 (Hint to Problem 3.62(b)). Replace twice \frac{1}{2} with (\frac{1}{2})^{\frac{1}{p}} ▷ p.192 – line 12 (Hint to Problem 3.62(b)). Replace k ≥1 with k ≥ m ▷ p.192 – last line (Hint to Problem 3.63). Replace m ∈ ℕ with n ∈ ℕ ▷ p.193 – line 3 (Hint to Problem 3.63). Replace V_{n}^{−} → 0 with diam(V_{n}^{−}) → 0 ▷ p.193 – line 11 (Problem 3.64(a)). Replace Problems with Problem ▷ p.194 – line 6 (Problem 3.66(b)). Replace

Problems 3.42(a), Theorems 3.40 and 3.81, with Corollary 3.81

Chapter 4

▷ p.198 – line 1. Replace incorporate with incorporates ▷ p.199 – line 1 (Definition 4.1). Replace x and y in X with x and y in X ▷ p.199 – line 23 (Proof of Proposition 4.2). Insert (iii) after “symmetry” ▷ p.204 – line 16. Replace 0 ≤ ∥αu∥ ≤ |α|∥u∥ = 0 with ∥αu∥ = |α|∥u∥ = 0 ▷ p.206 – line 5. Replace dicuss with discuss ▷ p.210 – line 12. Insert . at the end of the displayed line ▷ p.211 – line 9. Insert Similarly, αx ∈ M^{−} for every scalar α.

[ (see Proposition 3.27 again). Similarly, αx ∈ M^{−} for every scalar α. ] ▷ p.212 – line 3 from bottom. Replace ⋁(⋂_{γ∈Γ}M_{γ}) with ⋁(⋃_{γ∈Γ}M_{γ})

▷ p.218 – lines 1 and 2 (far above Example 4H). Delete ≤ |α|∥T ∥∥x∥ ▷ p.218 – line 4 (displayed). Replace ∥αT ∥ ≤ |α|∥T ∥ with ∥αT ∥ = |α|∥T ∥ ▷ p.222 – line 3. Replace 1≤ ∥P_{i}∥_{∞} and 1≤ ∥P_{i}∥_{∞} with 1≤ ∥P_{i}∥_{∞} and 1≤ ∥P_{i}∥_{p} ▷ p.222 – line 3 (Example 4I). Replace P_{k} ≠ O with P_{i} ≠ O ▷ p.224 – line 1. Insert T ∈ B[X , Y] and after “Moreover, if” ▷ p.225 – line 11 from bottom (above Theorem 4.21). Replace into with onto ▷ p.225 – line 6 from bottom (Proof of Theorem 4.21). Replace de with be ▷ p.226 – line 7 from bottom (Proof of Theorem 4.21). Replace T x_{n} with T x_{k} ▷ p.229 – line 11 from bottom (above Theorem 4.25). Replace if with of ▷ p.233 – line 9 (Proof of Theorem 4.27). Replace ∥a∥_{∞} with ∥a∥_{∞} = 1 ▷ p.235 – line 7 from bottom. Replace 4.30(b) with 4.30, part(b)) ▷ p.237 – line 10 (Proof of Corollary 4.34). Replace B_{1}[0] with B_{1}[0] ▷ p.238 – lines 5 and 4 from bottom. Replace twice W with Ŵ ▷ p.240 – line 6 (above Theorem 4.39). Replace arrises with arises ▷ p.240 – line 9 from bottom (Proof of Theorem 4.40). Replace M with N ▷ p.241 – line 9 from bottom (afterTheorem 4.41). Replace B[X̃, Ŷ] with B[X̃, Ỹ] ▷ p.243 – lines 14,16 (Proof of Theorem 4.43).Replace twice B_{ρ}(x_{0}) with B_{2ρ}(x_{0}) ▷ p.243 – line 16 (Proof of Theorem 4.43). Replace B_{ρ}(0) with B_{2ρ}(0) ▷ p.245 – line 4 (Proof of Proposition 4.46). Replace e_{i} ∈ B with e_{i} ∈ B ▷ p.245 – line 10 from bottom. Replace an operator with a transformation ▷ p.248 – line 4 (Example 4L). Replace S_{−} = with S_{−}x = ▷ p.250 – line 15 (Example 4M). Replace ∑_{k=0}^{∞}|ξ_{k}|^{k} with ∑_{k=0}^{∞}|ξ_{k}|^{p} ▷ p.251 – line 13 (Proof of Proposition 4.50). Replace 4.30 with 4.32 ▷ p.254 – line 5 from bottom (Example 4N).Replace Problem with Example ▷ p.255 – lines 9–12 (Example 4O). 4 lines with 3 displayed formulas should read:

B_{0}[X , Y]^{−} lies in B_{∞}[X , Y],

and

B_{∞}[X , Y] is closed in B[X , Y],

▷ p.255 – line 18 (Example 4O). Insert after “.” and before “Conclusion:”

Moreover, strong convergence to compact diagonals was shown in Problem 4.53. ▷ p.255 – line 20 (Example 4O). Replace which implies that with and ▷ p.256 – line 1 (Proof of Proposition 4.56). Replace 4.52(b) with 4.52(f) ▷ p.259 – line 9 from bottom (Proof of Lemma 4.59). Replace f(x) with f_{1}(x) ▷ p.262 – line 9 (Proof of Theorem 4.61). Replace ig(x) with ig(ix) ▷ p.267 – line 10 from bottom (Example 4Q). Replace 3Q with 4Q ▷ p.268 – line 18 (Example 4S, line 4). Replace space with spaces ▷ p.269 – line 19 (Problem 4.1(a)). Replace each α with each nonzero α ▷ p.270 – line 10 from bottom (Problem 4.3(iii’)). Replace α ≤ 1 with |α| ≤ 1 ▷ p.276 – line 10 from bottom (Problem 4.11). Replace ∑_{1=0}^{n} with ∑_{i=0}^{n} ▷ p.281 – line 15 (Problem 4.19). Replace if the identity with of the identity ▷ p.288 – line 2 (Problem 4.31). Replace L ∈ L[X , Y] with L ∈ L[M, Y] ▷ p.288 – line 11 (Hint to Problem 4.31). Insert ) after “ Y” and before “.” ▷ p.290 – line 14 (Problem 4.35). Replace Theorem 2.12 with Theorem 2.19 ▷ p.290 – line 4 from bottom (Problem 4.35). Replace ⊂ with ⊆ ▷ p.293 – last line (Problem 4.40). Replace This called with This is called ▷ p.294 – line 4 (displayed formula, Problem 4.41). Replace

∥T V ∥ = ∥T ∥ and ∥V S∥ = ∥S∥ with

∥V S∥ = ∥S∥ and ∥T V ∥ = ∥T ∥ if V is surjective

▷ p.294 – line 4 from bottom (Problem 4.43). Replace X with X ▷ p.296 – line 5 from bottom. Replace von Newman with Newman ▷ p.299 – line 12 (Problem 4.52). Replace B[ℓ_{+}^{p}] with B[ℓ_{+}^{p}]

Chapter 5

▷ p.312 – line 19 (above Definition 5.1). Replace v ∈ 𝔽 with v ∈ X ▷ p.314 – line 2 from bottom. Replace inner space with inner product space ▷ p.319 – line 9 (Example 5B). Replace ⟨ , ⟩ with ⟨ ; ⟩ ▷ p.319 – last line (Example 5C). Replace Example 3D with Example 4D ▷ p.321 – line 12 (Example 5D). Replace since every with for every ▷ p.324 – line 14 (displayed formula below Definition 5.7). Replace = with ⊆ ▷ p.326 – line 14 from bottom. Replace and with And ▷ p.326 – lines 3 and 2 from bottom. Replace three times

(M⟨ ; ⟩) with (M, ⟨ ; ⟩) and (M⟨ ; ⟩_{M}) with (M, ⟨ ; ⟩_{M}) ▷ p.327 – line 5. Replace Proposition 4.4 with Proposition 4.7 ▷ p.332 – line 5 (above Corollary 5.14). Replace in x in H. with x in H. ▷ p.332 – lines 12 and 5 from bottom. Replace twice M^{⊥⊥} with M^{⊥⊥} ▷ p.333 – line 16 (Example 5H, line 6). Replace {x}^{⊥} with {z}^{⊥} ▷ p.333 – line 25 (Example 5H, line 15). Replace scalar with nonzero scalar ▷ p.336 – line 4. Replace Corollary 5.10 with Corollary 5.11 ▷ p.339 – line 5 (Proof of Proposition 5.19). Replace v is N with v in N ▷ p.341 – line 1 (Statement of Theorem 5.23). Delete operator ▷ p.342 – line 1 (Proof of Proposition 5.24). Delete . after “equivalent” ▷ p.346 – line 8 (Proof of Corollary 5.28). Replace ∪ with ∩ ▷ p.347 – line 4 from bottom (displayed formula, line 2). Replace = with ≤ ▷ p.349 – line 10 (Proof of Proposition 5.31). Replace Suppose with suppose ▷ p.349 – line 10 (Proof of Proposition 5.31). Replace ∑_{N∈Γ} with ∑_{k∈N} ▷ p.349 – line 11 (Proof of Proposition 5.31). Replace < α − ε with > α − ε ▷ p.352 – line 17 (Proof of Proposition 5.33). Replace ⟨x_{0} ; x_{i}⟩ with ⟨x_{i}; x_{0}⟩ ▷ p.356 – line 20 (Proof of Theorem 5.42). Replace basis with bases ▷ p.357 – line 18 (Proof of Proposition 5.43). Replace an with a smallest ▷ p.359 – line 13 from bottom (Example 5.L). Replace basis with bases ▷ p.359 – line 4 from bottom (Example 5.L). Replace e_{k} is an with e_{k} is a ▷ p.360 – line 7 (Example 5.L). Replace e_{k} is an with e_{k} is a ▷ p.360 – line 10 (Example 5.L). Replace basis with bases ▷ p.360 – last line (displayed formula). Replace ε with ε^{2} ▷ p.361 – line 11 (Proof of Theorem 5.48). Replace If with if ▷ p.361 – line 14 (Proof of Theorem 5.48). Replace ∥e∥ with ∥e∥^{2} ▷ p.363 – line 18 (Example 5L(b)). Replace {ξ_{j}}_{l∈ℕ} with {ξ_{j}}_{j∈ℕ} ▷ p.367 – line 19 (Proposition 5.51(a)). Replace P ∈ B[H] with P ∈ B[X] ▷ p.368 – line 3 from bottom. Replace orthonormal with orthogonal ▷ p.369 – line 12. Replace such with such that ▷ p.370 – line 4 from bottom. Replace of next with of the next ▷ p.372 – line 13 (Example 5O). Replace orthonormal with orthogonal ▷ p.374 – line 20 (above Definition 5.60). Replace {R(P_{k})}_{k∈ℕ} with {P_{k}}_{k∈ℕ} ▷ p.376 – line 9 (Proof 5.61). Replace (c) implies (d) with (c) implies (b) ▷ p.378 – line 10 from bottom (Corollary 5.64). Replace ⟨ ⟩_{∗} with ⟨ ; ⟩_{∗} ▷ p.379 – line 21 (above Proposition 5.65). Replace |f(x)| with |f_{y}(x)| ▷ p.381 – line 24. Insert if X is a Hilbert space, before “which” ▷ p.386 – line 25 (Proof of Theorem 5.70, line 2). Insert the after “of” ▷ p.387 – line 8 from bottom. Replace Theorem 5.67 with Proposition 5.67 ▷ p.388 – line 10 from bottom (displayed formula). Replace (S^{∗}+T^{∗}) with (S+T)^{∗} ▷ p.389 – line 14 (Example 5P, below displayed matrix). Replace υ_{n} with υ_{m} ▷ p.391 – line 14 (Prop. 5.72). Replace An operator with A transformation ▷ p.394 – line 16 (Proof of Proposition 5.77). Replace R(T) ⊆ with R(T) ⊆ ▷ p.395 – line 3 (Proof of Proposition 5.77). Replace 5.73(b^{∗}) with 5.76(b^{∗}) ▷ p.395 – line 4 from bottom. Replace R(T T^{∗})^{−} with R(T T^{∗})^{−} ▷ p.396 – line 3 from bottom (Proposition 5.78). Replace ⟨T x; x⟩ with |⟨T x; x⟩| ▷ p.396 – last line (displayed formula). Replace |⟨T u; u⟩ with |⟨T u; u⟩| ▷ p.399 – lines 1 and 2. Replace all T with P (five times) ▷ p.400 – line 13 (above Example 5R). Replace 5.76(b*) with 5.76(a*) ▷ p.402 – line 2 from bottom (Proof of Theorem 5.85). Correct: B_{2} − B_{1} = \frac{1}{8}R^{2} ▷ p.404 – line 3 from bottom. Replace R(P) with R(P) ▷ p.405 – lines 11,12 (Proposition 5.88). Insert . twice at the end of the lines ▷ p.406 – line 17 (Theorem 5.89). Replace Corollary 5.75 with Proposition 5.76 ▷ p.407 – line 1 (Proof of Theorem 5.89). Replace 4.30 with 4.38 ▷ p.407 – line 15 (Proof of Theorem 5.89, displayed formula).Replace H with H ▷ p.407 – line 10 from bottom. Replace R((T^{∗}T)^{−} with R(T^{∗}T)^{−} ▷ p.410 – last line (Problem 5.5). Replace |x_{n}(x)| with |x_{n}(y)| ▷ p.411 – lines 21, 22 (Problem 5.6). Replace second x_{1}(n) x_{1} with x_{2}(n) x_{2} ▷ p.412 – line 8 (Problem 5.8). Replace Proposition with Propositions ▷ p.415 – line 2 (Problem 5.13). Replace (1 − \frac{1}{k^{2}} )^{2} with (1 − \frac{1}{k^{2}} )^{\frac{1}{2}} ▷ p.416 – line 25 (Problem 5.17). Delete and ▷ p.416 – line 26 (Problem 5.17). Replace orthogonal with orthonormal ▷ p.419 – lines 3 and 4 (Problem 5.22). Swap converge with converges ▷ p.419 – lines 13 and 15 (Problem 5.23). Replace twice ∥T^{n}∥ with ∥T_{n}∥ ▷ p.419 – line 14 (Problem 5.23). Replace Problems 5.5 with Problem 5.5 ▷ p.421 – line 11 (Problem 5.28). Replace B[H_{j}, H_{i}] with B[H_{j}, K_{i}] ▷ p.421 – line 6 from bottom (Problem 5.29). Replace K_{k+1} with H_{k+1} ▷ p.422 – line 6. Replace matrix of operators with operator matrices ▷ p.422 – last line (Problem 5.30). Replace K_{k+1} with H_{k+1} ▷ p.423 – line 11. Replace matrix of operators with operator matrices ▷ p.423 – line 4 from bottom (Problem 5.30(c)).Replace unilateral with bilateral ▷ p.423 – line 3 from bottom (Problem 5.30(c)). Replace ⨁_{k=∞}^{∞} with ⨁_{k=−∞}^{∞} ▷ p.423 – line 2 from bottom (Problem 5.30(c)). Replace ⨁_{k−∞}^{∞} with ⨁_{k=−∞}^{∞} ▷ p.423 – last line and p.424 – lines 2–5 (Hint to Problem 5.30(c)). Replace all

∑k=−∞^{–1} with ∑k=−∞^{–n/2 –1} and all ∑_{k=0}^{∞} with ∑_{k= –n/2}^{∞}

▷ p.424 – line 3. Replace ( ∑k=−∞^{–1} ∥x_{k+n}∥^{2})^{\frac{1}{2}} ∥x∥ with ( ∑k=−∞^{–n/2 –1} ∥x_{k}∥^{2})^{\frac{1}{2}} ∥x∥

▷ p.424 – line 5. Replace lim_{n} ∑k=−∞^{–1} ∥x_{k+n}∥^{2} with lim_{n} ∑k=−∞^{–n/2 –1} ∥x_{k}∥^{2}

▷ p.425 – line 5 from botom (Problem 5.32(c) line 3). Delete n ▷ p.426 – line 2 (Problem 5.33). Replace Example 3L(c) with Example 5L(c) ▷ p.430 – line 12. Replace ≤ |⟨(Q_{n} − Q)x; x⟩| with = ⟨(Q_{n} − Q)x; x⟩ ▷ p.431 – line 19 (Hints to Problem 5.53). Replace 5.22(d) with 5.52(d) ▷ p.434 – lines 9 and 7 from bottom. Replace twice −Q^{1/2}))x∥ with −Q^{1/2})x∥ ▷ p.436 – line 8 from bottom (Hint to Problem 5.66(a)). Replace = with ≤ ▷ p.437 – line 6 from bottom. Replace {⟨T e_{γ}; e_{γ}⟩} with {⟨T e_{γ}; e_{γ}⟩}_{γ∈Γ}

Chapter 6

▷ p.443 – line 2 should read: (c)⇔(d) by Propositions 4.37 and 6.1. ▷ p.443 – line 20 (Proposition 6.4). Replace U: N^{⊥} → H is unitary. with

U: N^{⊥} → N^{⊥} ⊆ H is unitary where N^{⊥} is U-invariant. ▷ p.443 – line 24 (Proof of Proposition 6.4). Replace = T T^{∗}T with = T^{∗}T T ▷ p.444, lines 17–20 (Proof of Proposition 6.4). Replace as follows.

Since T is normal, T commutes with T^{∗} and T. Hence N = N (T) reduces T (Problem 5.34) and so does N^{⊥}. Then N^{⊥} is V -invariant because T = V P Q, and so the above inclusion holds for I and V V^{∗} in B[N^{⊥}].

▷ p.447 – line 4 (Proof of Lemma 6.8). Replace p_{n}, q_{n} ∈ ℕ with p_{n}, q_{n} ∈ ℕ_{0} ▷ p.447 – lines 5 and 4 from bottom. Replace twice every with for every ▷ p.449 – line 8 from bottom (Corollary 6.11). Replace d(λ − ς) with d(λ , ς) ▷ p.449 – line 4 from bottom. Replace von Newman with Newman ▷ p.450 – line 7 (displayed formula, above Proposition 6.13). Replace R with R ▷ p.450 – line 9 from bottom (Proposition 6.13). Insert If σ(T) is empty, then

[ Claim 1. If σ(T) is empty, then φ ∘ R : ρ(T) → ℂ is bounded. ] ▷ p.450 – line 5 from bottom. Insert if σ(T) is empty after “Theorem 3.86” ▷ p.452 – Diagram. — First row: Insert vertical line between R_{λ}^{−}= X and R_{λ}^{−}≠ X

— Forth row: Replace T_{λ}^{−1}∈ B[R_{λ},X] with T_{λ}^{−1}∉ B[R_{λ},X]

▷ p.452 – line 10 from bottom (Proposition 6.14). Replace {α_{k}}_{i=1}^{n} with {α_{i}}_{i=1}^{n} ▷ p.453 – line 9 from bottom (Proof of Proposition 6.16). Replace 3.32 with 3.27 ▷ p.454 – lines 11 and 12 (Proof of Proposition 6.16). Replace as follows.

Moreover, since λ ∈ ℂ\σ_{AP} (T) if and only if λI − T is bounded below, the same argument of Proposition 6.11 shows that ℂ\σ_{AP} (T) is open. ⊓ ⊔

▷ p.454 – lines 7 and 5 from bottom. Replace twice spectrum with spectra ▷ p.455 – line 13 (Proof of Proposition 6.17). Replace σ_{AP} (T^{∗}) with σ_{AP} (T^{∗})^{∗} ▷ p.456 – line 12 from bottom. Replace σ(R) ∈ [0, 1] with σ(R) ⊆ [0, ∞) ▷ p.456 – lines 9, 8, 6 from bottom. Replace three times σ_{P} (T) with σ_{P} (P) ▷ p.456 – line 4 from bottom. Replace ρ(T) with ρ(P) ▷ p.457 – line 3. Replace σ_{P} (T) and ρ(T) with σ_{P} (J) and ρ(J) ▷ pp.457,458 (Proof of Theorem 6.19) Replace nine times α_{n}^{−1} with (−1)^{n+1}α_{n} ▷ p.458 – line 3 (Proof of Theorem 6.19). Replace μ ∈ ρ(T) with μ ∈ ρ(p(T)) ▷ p.460 – line 11. Replace von Newman with Newman ▷ p.460 – line 10 from below (Proof of Proposition 6.21). Replace R with R ▷ p.461 – line 10 (above Proposition 6.22). Delete the ▷ p.462 – line 6. Replace von Newman with Newman ▷ p.462 – line 11 (Corollary 6.23). Delete , and ∥(λI − T)^{−1}∥ ≤ (|λ| − ∥T ∥)^{−1} ▷ p.463 – line 24 (Proof of Proposition 6.24). Insert , after “ that is” ▷ p.464 – line 16 (Proof of Proposition 6.25). Replace W (T) with W (T)^{−} ▷ p.466 – lines 4 and 5. Replace ∥T ∥ = ⟨T x; x⟩ with ∥T ∥ = |⟨T x; x⟩| ▷ p.466 – line 23 (Example 6B). Replace T ∈ B[H] with T ∈ B[X] ▷ p.466 – line 24 (Example 6B). Replace manifolds with linear manifolds ▷ p.467 – line 15 (Example 6C). Replace σ_{R}(T^{∗})^{∗} with σ_{P} (T^{∗})^{∗} ▷ p.469 – line 10 (Example 6E – displayed). Replace ⨁_{k=∞}^{∞} with ⨁_{k=−∞}^{∞} ▷ p.470 – line 11 (Example 6C). Replace ({α_{k}}_{k=−∞}^{∞}) with shift({α_{k}}_{k=−∞}^{∞}) ▷ p.470 – last line (Example 6F). Replace ∥y_{k}∥^{2} with ∥y_{k+1}∥^{2} ▷ p.471 – line 6 from bottom (Example 6G). Replace ⊆ with ∈ ▷ p.472 – line 8 from bottom (Example 6G). Replace 0 ≠ σ_{R}(T) with 0 ∉ σ_{R}(T) ▷ p.473 – line 2. Replace matrix of operators with operator matrices ▷ p.473 – line 3 from bottom (Example 6H). Replace λ ≠ σ_{P} (T) with λ ∉ σ_{P} (T) ▷ p.474 – line 12 (Example 6H). Replace Example 6E with Example 6D ▷ p.474 – line 15 (Example 6H – displayed formula). Swap σ_{R}(T) with σ_{P} (T) ▷ p.474 – lines 19,20 (Example 6H). Replace

is not included in the boundary of the spectrum with has nonempty interior ▷ p.477 – line 12 (displayed). Replace at the end of the line , with . ▷ p.477 – line 9 from bottom. Replace Theorem 3.30 with Corollary 6.31 ▷ p.478 – lines 19, 20, 21 (Proof of Proposition 6.33). Replace as follows.

compact, then Theorem 4.52 ensures that{λ_{n}^{−1}y_{n}}has no bounded subsequence. That is, sup_{k}|λ_{k}|^{−1} = sup_{k}∥λ_{k}^{−1}y_{k}∥ = ∞, and so inf_{k}|λ_{k}| = 0 for every subsequence {λ_{k}} of {λ_{n}}. Thus λ_{n} → 0. ⊓ ⊔

▷ p.479 – line 9 from bottom. Replace exists with exist ▷ p.479 – line 7 from bottom. Replace basis with bases ▷ p.481 – line 13 (Proposition 6.36). Replace ⟨P_{γ} x; P_{γ} x⟩ with ⟨P_{γ} x; P_{γ} y⟩ ▷ p.481 – line 2 from bottom (displayed). Replace λ ∈ Γ with γ ∈ Γ ▷ p.481 – line 2 from bottom (displayed). Replace σ_{R}(T) ≠ ∅ with σ_{R}(T) = ∅ ▷ p.481 – last line (displayed). Replace λ ∈ Γ with γ ∈ Γ ▷ p.482 – lines 1 and 3 (Proposition 6.37). Replace twice ∑_{γΓ} with ∑_{γ∈Γ} ▷ p.482 – line 21 (Proposition 6.37). Insert x after “P_{β}” and after “P γ” ▷ p.482 – line 27 (Proposition 6.37). Replace ρ(T)\σ_{P} (T) with (ℂ\ρ(T))\σ_{P} (T) ▷ p.483 – line 8 from bottom. Replace Definition 4.45 with Corollary 4.55 ▷ p.487 – line 7 from bottom. Replace Corollary 6.40 with Proposition 6.40 ▷ p.490 – line 4 from bottom. Replace Since with since ▷ p.491 – last line. Replace Nikodýn with Nikodým ▷ p.492 – line 5 (Theorem 6.47, displayed formula). Replace , with . ▷ p.499 – line 6 from bottom. Delete , at the end of the line ▷ p.492 – line 13 (below Theorem 6.48). Replace R(P (Λ)) with R(P (Λ)) ▷ p.501 – line 12. Replace H ≠ {0} with H of dimension greater than one ▷ p.502 – line 3 from bottom. Replace XX^{∗} with X^{∗}X + Y^{∗}Y − Y Y^{∗} ▷ p.502 – line 2 from bottom (Problem 6.16). Replace X^{∗}x with X^{∗}u ▷ p.503 – line 2 from bottom. Replace Problem 6.17 with the above result ▷ p.504 – line 7 from bottom (Problem 6.23). Replace M^{⊥} = ∅ with M^{⊥} = {0} ▷ p.506 – line 5 from bottom. Replace , with . before “Moreover” ▷ p.507 – lines 2–14 (Hint to Problem 6.27(b)). Replace as follows.

Hint: Take any 0 ≠ α ∈ ℝ. Since αi ∈ ρ(T) and (αiI − T) is hyponormal, ∥(αi − T)^{−1}∥ ≤ |α|^{−1} (Problem 6.14). Hence α^{2}∥(αiI − T)^{−1}(αiI − T)x∥^{2} ≤ α^{2}∥x∥^{2}+∥T x∥^{2}−2Re⟨αix; T x⟩. Thus 2αIm⟨T x; x⟩ ≤ ∥T x∥^{2} and Im⟨T x; x⟩ = 0.

References & Index

▷ p.511 – line 12 from bottom. Replace Notes in with Notes on ▷ p.513 – line 13. Replace Rolt with Holt ▷ p.515 – last line. Replace 6nd with 6th ▷ p.518 – line 23, right column. Replace comeagre with comeager ▷ p.522 – line 23, right column. Replace meagre with meager ▷ p.523 – line 7, left column. Replace nonmeagre with nonmeager ▷ p.524 – line 5 from bottom, right column. Replace Nikod’yn with Nikodým ▷ p.525 – line 20, left column. Delete restriction of a function, 5 ▷ p.527 – lines 6 and 7 from bottom, left column. Insert page number 412

(November 12, 2017)