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1 | (10) |
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1.1 What is Quaternionic Spectral Theory? |
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1 | (5) |
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1.2 Some Historical Remarks on the S-Spectrum |
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6 | (3) |
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1.2.1 The Discovery of the S-Spectrum |
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6 | (2) |
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1.2.2 Why Did It Take So Long to Understand the S-Spectrum? |
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8 | (1) |
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1.3 The Fueter--Sce--Qian theorem and spectral theories |
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9 | (2) |
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2 Slice Hyperholomorphic Functions |
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11 | (42) |
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2.1 Slice Hyperholomorphic Functions |
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13 | (20) |
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2.2 The Fueter Mapping Theorem in Integral Form |
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33 | (5) |
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2.3 Vector-Valued Slice Hyperholomorphic Functions |
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38 | (10) |
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48 | (5) |
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3 The S-Spectrum and the S-Functional Calculus |
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53 | (22) |
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3.1 The S-Spectrum and the S-Resolvent Operators |
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53 | (10) |
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3.2 Definition of the S-Functional Calculus |
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63 | (6) |
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69 | (6) |
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3.3.1 The Left Spectrum σL(T) and the Left Resolvent Operator |
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69 | (2) |
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3.3.2 Power Series Expansions and the S-Resolvent Equation |
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71 | (4) |
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4 Properties of the S-Functional Calculus for Bounded Operators |
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75 | (50) |
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4.1 Algebraic Properties and Riesz Projectors |
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75 | (7) |
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4.2 The Spectral Mapping Theorem and the Composition Rule |
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82 | (5) |
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4.3 Convergence in the S-Resolvent Sense |
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87 | (3) |
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4.4 The Taylor Formula for the S-Functional Calculus |
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90 | (14) |
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4.5 Bounded Operators with Commuting Components |
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104 | (4) |
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4.6 Perturbations of the SC-Resolvent Operators |
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108 | (4) |
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112 | (4) |
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116 | (9) |
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4.8.1 The S-Functional Calculus for n-Tuples of Operators |
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117 | (3) |
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4.8.2 The W-Functional Calculus for Quaternionic Operators |
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120 | (5) |
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5 The S-Functional Calculus for Unbounded Operators |
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125 | (12) |
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5.1 The S-Spectrum and the S-Resolvent Operators |
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126 | (4) |
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5.2 Definition of the S-Functional Calculus |
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130 | (6) |
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136 | (1) |
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6 The H∞-Functional Calculus |
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137 | (14) |
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6.1 The Rational Functional Calculus |
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137 | (2) |
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6.2 The S-Functional Calculus for Operators of Type ω |
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139 | (3) |
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6.3 The H∞-Functional Calculus |
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142 | (2) |
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6.4 Boundedness of the H∞-Functional Calculus |
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144 | (2) |
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146 | (5) |
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6.5.1 Comments on Fractional Diffusion Processes |
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146 | (5) |
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7 The F-Functional Calculus for Bounded Operators |
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151 | (26) |
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7.1 The F-Resolvent Operators and the F-Functional Calculus |
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151 | (8) |
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7.2 Bounded Perturbations of the F-Resolvent |
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159 | (4) |
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7.3 The F-Resolvent Equations |
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163 | (2) |
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7.4 The Riesz Projectors for the F-Functional Calculus |
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165 | (3) |
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7.5 The Cauchy-Fueter Functional Calculus |
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168 | (3) |
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171 | (6) |
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7.6.1 The F-Functional Calculus for n-Tuples of Operators |
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172 | (2) |
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7.6.2 The Inverse Fueter--Sce Mapping Theorem |
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174 | (3) |
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8 The F-Functional Calculus for Unbounded Operators |
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177 | (10) |
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8.1 Relations Between F-Resolvent Operators |
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178 | (3) |
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8.2 The F-Functional Calculus for Unbounded Operators |
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181 | (2) |
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183 | (4) |
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8.3.1 F-Functional Calculus for n-Tuples of Unbounded Operators |
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184 | (3) |
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9 Quaternionic Operators on a Hilbert Space |
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187 | (32) |
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187 | (5) |
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9.2 The S-Spectrum of Some Classes of Operators |
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192 | (4) |
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9.3 The Splitting of a Normal Operator and Consequences |
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196 | (8) |
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9.4 The Continuous Functional Calculus |
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204 | (13) |
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217 | (2) |
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219 | (14) |
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10.1 Spectral Integrals for Bounded Measurable Functions |
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220 | (5) |
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10.2 Spectral Integrals for Unbounded Measurable Functions |
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225 | (6) |
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10.3 Comments and remarks |
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231 | (2) |
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11 The Spectral Theorem for Bounded Normal Operators |
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233 | (12) |
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11.1 Construction of the Spectral Measure |
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234 | (7) |
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11.2 The Spectral Theorem and Some Consequences |
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241 | (2) |
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11.3 Comments and Remarks |
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243 | (2) |
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12 The Spectral Theorem for Unbounded Normal Operators |
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245 | (10) |
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12.1 Some Transformations of Operators |
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245 | (2) |
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12.2 The Spectral Theorem for Unbounded Normal Operators |
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247 | (3) |
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12.3 Some Consequences of the Spectral Theorem |
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250 | (3) |
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12.4 Comments and Remarks |
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253 | (2) |
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13 Spectral Theorem for Unitary Operators |
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255 | (18) |
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13.1 Herglotz's Theorem in the Quaternionic Setting |
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255 | (4) |
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13.2 Preliminaries for the Spectral Resolution |
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259 | (6) |
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13.3 Further Properties of Quaternionic Riesz Projectors |
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265 | (3) |
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13.4 The Spectral Resolution |
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268 | (3) |
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13.5 Comments and Remarks |
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271 | (2) |
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14 Spectral Integration in the Quaternionic Setting |
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273 | (30) |
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14.1 Spectral Integrals of Real-Valued Slice Functions |
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274 | (4) |
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278 | (7) |
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14.3 Spectral Systems and Spectral Integrals of Intrinsic Slice Functions |
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285 | (10) |
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14.4 On the Different Approaches to Spectral Integration |
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295 | (8) |
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15 Bounded Quaternionic Spectral Operators |
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303 | (34) |
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15.1 The Spectral Decomposition of a Spectral Operator |
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303 | (21) |
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15.2 Canonical Reduction and Intrinsic S-Functional Calculus for Quaternionic Spectral Operators |
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324 | (13) |
| Contents of the Monograph: Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes |
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337 | (2) |
| Index |
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339 | (4) |
| Bibliography |
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343 | |
Lisainfo e-raamatute kohta