Fattorini H.O., Kerber A. The Cauchy Problem

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Notation and Subject Index

4 -(closure of an operator), 13

A*, see Adjoint (of an operator)

A" (Phillips adjoint), 79

Ao, 125, 127, 199, 200

A, 206

(

A)* (fractional power of an operator),

360

AF, 307

A,, (extended limit of a sequence

of operators), 322

A(#), 135, 218

Ao(/3), 220

A,($), 142, 236

236

A($o,/3,), 143

146

A(t), 428

(0, A), 94

(1, A), 94

(T, 176

d(9p), ?(4p-), 176

A-admissible Banach spaces, 307

Abel mean, 94

Abel-Poisson semigroup, 115

Absolute value (of an element in a Banach

lattice), 160

Abstract Cauchy problem, see Cauchy

problem

Abstract elliptic operators, 173

Abstract Gevrey spaces, 504

Abstract Goursat problem, 108

Abstract parabolic distributions, 493-496

Abstract parabolic equations, 172-187;

time-dependent, 390-419

Accretive operators, 167

Addition of infinitesimal generators, 338-340

Addition problem, 338

Adjoint

of bounded operators, 4

formal, see Formal adjoints

Hilbert space, 16

of operators in (2, C, 77-80

of ordinary differential operators, 129,

134-135,138,142,146

of elliptic partial differential operators,

221, 235-237

Phillips, see Phillips adjoint

of symmetric hyperbolic operators, 148

of unbounded operators, 5-14, 16

of vector-valued distributions, 500

Adjoint boundary condition

for ordinary differential operators, 138

627

628

for elliptic partial differential operators,

213

Adjoint equation, 77-80

Alaoglu's theorem, 15

Analytic continuation of a semigroup,

176-179

Analytic functions (with values in a Banach

space), 5-6

Analytic semigroups, 176-187

Approximate resolvent, 103

Approximation of abstract differential

equations, 316-336

Approximation of Banach spaces, 317

Approximation theory and semigroup theory,

85, 115-116

Associated map (of a boundary patch), 205

Bx(u, v), 216

BA,fl(u,v), 217

BC(-oo,oo), 348

BC (N )(R--'), 58

/3, $o, /3,, see boundary condition

/3', see adjoint boundary condition

Banach lattices, 160-166

Banach spaces, I

A-admissible, 307

approximation of, 317

reflexive, 5, 125

separable, 15

Banach-Steinhaus theorem, I

Bessel potential operator, 311

Boundary condition

for ordinary differential operators, 135,

142

for elliptic partial differential operators,

207, 215

of types (1), (II), 135, 142, 215

Boundary patch V°, 205

associated map, 205

Boundary value problem

for abstract differential equations, 378

for the wave equation, 378-380

= (bounded convergence of vector-valued

distributions), 468

Bounded operator, I

Brownian motion, 95

Co, 94

C(K),

Co(K),

21-22

C(K)*, CO(K)*,

21-22

C(k)(ff), C(°o)(2), C(k) 23

C(K)R, 22

C(k) (domain of class), 204

Notation and Subject Index

C(k'(12) (operator of class), 223

Co(-oo,oo), 126

CB[O,oo), Co,0[0,oo), 135

C [0,1 ], Co [0,1 ], C, [0,1 ], Co,, [0, 1], 143

236

z)(Q)fl, 220, 235

C(')(0, T; E), 59

Cam, a, 348

CT(A*), 89

Cj )), 56

(0, CI), 94

(1, C,), 94

C v-dimensional unitary space

L+, (2+(w), (2,(C, w), (2, L'(w), (2(C, w),

72-73

C'+(C, co; E), C'+(E), 306

C°°, 174-175

C (°°) ((0, oo); ( E; X))-',

493

Cauchy data, 54

Cauchy-Poisson semigroup, 1 16

Cauchy-Polya theorem, 80

Cauchy problem (or initial value problem)

abstract, in t

' 0, 29-32, 63

abstract, in - oo < t 5 cc, 43, 72

abstract, for time-dependent equations,

382

incomplete, 352-354, 372-374, 378

in the sense of Hadamard, 55

in the sense of vector-valued distributions,

482

mildly well posed, 498

reversed, for abstract parabolic equations,

354-357

Cauchy-Schwarz inequality, 4

Causality (of a system), 479

Cesaro mean, 94

( )` (convex hull), 469

Chapman-Kolmogoroff equation, 96

Characteristic function (of an operator-valued

distribution), 485

Closable operators, 13

Closed graph theorem

for Banach spaces, 7

for Frechet spaces, 56

Closed operators, 7

Closure of an operator, 13

Complex vs. real spaces, 123-125

Conormal derivative, 206-207

Invariance of, 209

weak, 231, 235

Conormal vector, 207

(convergence in a sequence of approximat-

ing Banach spaces), 316

Notation and Subject Index

Conservation of norm (of solutions of

abstract differential equations)

in t > 0, 155-156, 159

in t > 0 for positive solutions, 165-166

in -oo<t<oo, 121-122, 159

Conservative operators, 154-156

Convolution

equations, 482

of functions, 23

of vector-valued functions, 269, 469

of vector-valued distributions, 468-470

inverses, 480, 483

Cosine functional equation(s), 99

Cosine functions

in Banach spaces, 99-101

infinitesimal generator, 100

in linear topological spaces, 103-104

representation (by groups), 100

Coulomb potential

for a particle, 293, 296

for a system of n particles, 294

Cover, 205

partition of unity subordinate to a cover,

205

Crossover of initial conditions in singular

perturbation, 343

( )a, 482

D (subspace of initial data), 29, 382

Dk = dldxk

Ds (derivative of fractional order), 477

D' (conormal derivative), 206-207

D(x, v), 274

D(A°°) (denseness of), 71, 501

6(S2),6

6D(k)(f2),6(k)(q),6

(9),23-24

6D,, 56

6D,, 89

%(K), 465

l '(Q; E) 6t '(E), 462

6o(Q, E) 6l o(E), 464

6D', ( E), 464

6 '((E, X)), 6D'((X; E)), 477, 480

6DO' (X; E)), 6D ()((E, X)), 483

6 +((E, X)), 477

6lo((-oo,a);

(E, X)),

484

6o((X; E))n 6lo((E; X))-', D ((E; X))-',

6i o n 6D L-', 'Dp-', 485

8-sequence, 23

Deficiency indices d, d-, 19

Delay differential equations, 487

Densely defined operator, 7

Difference-differential equations, 487

Diffusion equation, 32-39

629

Dirac equation

for a free particle, 52-53

for a particle, 296-297

perturbation theory of, 296-297

Dirichlet boundary condition (or boundary

condition of type (II)), 215

Discrete semigroups, 329

generator, 329

Dispersive operators, 162-165

Dissipative operators, 120-125, 154-159

closure of, 122

extensions of, 122-123, 156-159

generation properties of, 121-122

in Hilbert space, 154-159

in real spaces, 124-125

with respect to a duality map, 120-125

Dirichlet kernel of the half-space, 254

Discrete semigroups, 329

Distribution semigroups

infinitesimal generator, 505

with values in a Banach space, 505

with values in a linear topological space,

506

Domain (of an operator), 6

Dual space

of a Banach space, 4

of a Hilbert space, 16

of LP(S2; µ) spaces, 20

of C(K), Cj(K ), Co (K) spaces, 21-22

Duality maps, 120

proper (in a Banach lattice), 161

proper (in C(K), C0(K), LP(Q, p)),

161-162

Duality sets, 118

in LP(f ), 119

in C(K), Cj(K), C0(K), 120

E*, see dual space

E+ (nonnegative elements of a Banach

lattice), 161

(E; F), (E), I

(E,,, 317

&'(E), 6;o(E), 464

&o((X; E)), 494

r1(map associated to a boundary patch), 205

Eigenspaces, 10

generalized, 10

Eigenvalues, 10

Eigenvectors, 10

generalized, 10

Elliptic (second order) partial differential

operators

630

analyticity of semigroups generated by,

255-258, 261-263

assignation of boundary conditions in

LP(R), 206-208; in C(S2), 209-213

of class C(k), 223

compactness of semigroups generated by,

263-264

degenerate, 221-222

in divergence form, 199

formal adjoint, 203

formally self adjoint, 203

m-dissipative extensions in L2(Q), 214-222

m-dissipative extensions in LP(Q),

235-236, 243, 255-258

m-dissipative extensions in L' (T2), 235-236,

251, 258-260

m-dissipative extensions in C(S2), Cp(Q),

235-236, 244, 260-261

perturbation of, 304-305

regularity theorems for, 223-235

transformation under change of variables,

208-209

variational form, 199

Ergodic theory, 97-98

E-valued distributions: see vector-valued

distributions

Evolution equations, 455

Evolution function, 95

Evolution operator, see Propagator

Existence (of solutions of an abstract

differential equation), 104

exp(tA ), 92

Explosive solutions

of abstract differential equations, 349

of the heat equation, 348-349

Exponential equation(s), 62-63

Exponential formulas, 92, 506

Extended limit of a sequence of operators

(ex-lim

322

f * g (convolution of functions), 23, 269, 469

f(t), 462

F- E, 306

9, 6S-', see Fourier transform

Faedo-Galerkin method, 460

Finite difference approximations to abstract

differential equations, 329-336

Finite difference schemes

explicit, 334-335

implicit, 335-336

Flow, 95

Focusing of waves, 49

Notation and Subject Index

Fokker-Planck equation, 98

Formal adjoint

of ordinary differential operators, 129

of elliptic partial differential operators,

203

of symmetric hyperbolic operators, 147

Fourier transform (or Fourier-Plancherel

transform), 24

Fractional derivatives, 477

Fractional powers of operators, 357-372,

375-378.

Friedrichs' lemma, 167

Functional calculus

for bounded operators, II

for unbounded operators, I I

for normal operators, 18

Functional differential equations, 507

linear autonomous, 508

y, yo, y,, see Boundary conditions

y', see Adjoint boundary condition

F(A) (graph of an operator), 7

F', F2, F3, 208

Generalized eigenspaces, 10

Generalized eigenvectors, 10

Generalized solution (of an abstract

differential equation), 30, 382

Generalized sequences, 15

Geometrical optics, 49

Gevrey spaces (abstract), 504

Goursat problem (abstract), 108-109

Graph of an operator, 7

Greatest lower bound (of two elements in

a Banach lattice), 160

Green function, 144-145

Green distribution (of a system), 480

Gronwall's lemma, 384

Groups

isometric, 121-122

isometric in Hilbert space, 159

strongly continuous, 83

strongly measurable, 86

translation, 74-75

uniformly continuous, 84

unitary, 159

HJ(f2), Hi, 24

H'(R"') H',

H'(R'")o, 25

Ho(?), 215

`3C(x; u) (Hessian matrix), 200

Hadamard's formulation of the Cauchy

problem 54-58, 503

Notation and Subject Index

Heat-diffusion equation, 27-29, 32-39,

348-354

analyticity of solutions, 37

compactness of solution operator, 37

conservation of L'-norm of nonnegative

solutions, 38

explosive solutions, 348

in Ce(ll), 27-29

in LP(Q), 32-39

nonuniqueness of solutions, 349

reversed Cauchy problem, 350-354

Hereditary differential equations, 507

Hessian matrix, 200

Higher order abstract differential equations,

98-102

reduction to first order, 98, 101, 103

Hilbert spaces, I

Hilbert space adjoints, 16

Hille-Yosida theorem, 82-83, 91

Hille approximation, 92

Hyperfunctions, 505

Histories (spaces of), 508-509

Huygens' principle, 94-95, 507

l (space of inputs of a system), 479

(x) (incoming vectors for neutron transport

equation), 273

Inclusion of operators, 14

Incomplete Cauchy problem

for the Laplace equation, 352-354

for abstract differential equations, 372-374

Infinitesimal generator

of a semigroup, 81

of a distribution semigroup, 505

of a cosine function, 100

Inhomogeneous abstract differential

equations

time-independent, 86-91, 111-112

time-dependent, 415-419, 452-454

Impulsive excitation, 480

Input (of a system), 479

Ihtegral equations (for construction of the

propagator), 391

Integro-differential equations, 507

Integration (of functions with values in a

Banach space), 3-4

Interior patch V', 205

Invertible operators, 2

Joint graph norm, 486

J (mollifier operators), 23

LP(l2 A), LP(Q), LP(ll: µ)', LP(Q)°, 20

LP(K;µ)c, LP(12;µ)a, 22

631

LP(Q; µ)", 20

L P(2), 25

LP(F), 219

L2(IX'), L2(R

), 25

LP(Q)p, LP(Q)p, 258-259

LP(0,1)p, LP(0,1)p, 193

1f(X) (Laplace transform of a vector-valued

function), 31

LU(A) (Laplace transform of a vector-valued

distribution), 473

A( a, ft), A (logarithmic region), 487

Lagrange identity, 129, 138, 148, 203, 213

Lagrange formula, 87

Laplace transform

of vector-valued functions, 31

of vector-valued distributions, 473

Lattice, 160

Lax equivalence theorem, 337

Lax-Milgram lemma, 214, 460

Least upper bound (of two elements in a

Banach lattice), 160

Left (product) integral, 456

Lie's exponential formula, 338

Limit circle, 168

Limit point, 168

Linear functionals, 4

Logarithmic convexity, 375-376

Logarithmic region, 487

Lyapunov's theorem, 166

m (u), 126, 200

m-dissipative operators, 123

Markov processes, 95-98

Chapman-Kolmogorov equation, 96

Fokker-Planck equation, 98

temporally homogeneous, 96

transition probability, 96

Maximal dissipative operators, 156-158

Maximal symmetric operators, 19, 158-159

Maxwell equations, 43-49

conservation of norm of solutions, 45

focusing of waves, 49

in L2(i83)6, 44-45

in LP(R3)6, 45-48

in Co(R3)6 48-49

time-reversibility, 45

Method of lines, 325

Mollifiers, 23

Mollifying sequence, 23

Mild solution (of an abstract differential

equation), 88

Modulus (of an element in a Banach lattice),

160

632

Multiplicity, 10

generalized, 10

v (outer normal vector), 204

v (conormal vector), 207

Negative part (of an element in a Banach

lattice), 160

Nets, 15

Neumann kernel of the half-space, 253

Neutron transport equation, 272-286

in LP(2), 273-280

in C(U), 280-286

incoming vectors, 273

spectrum of neutron transport operator,

286

velocity groups, 286

Nonlinear equations, 170-171

Nonlinear semigroups, 170-171

Nonnegative elements in a Banach lattice,

161

Normal operator, 16

0 (space of outputs of a system), 479

12(a, f6) (reverse logarithmic region), 494

Open map theorem, 7

Operational differential equations, 459

Operators (see Ordinary differential

operators, Partial differential operators)

accretive, 167

bounded,l

closable, 13

closed, 7

conservative, 154-156

dispersive, 162-165

dissipative, 120-125, 154-159

dissipative with respect to a duality map,

120-125

dissipative in Hilbert space, 154-159

normal, 16

positive, 162

resolvent, 2, 7-9, 14

resolvent set, 2, 7, 14

self adjoint, 17

spectral radius, 6

spectrum, 2

unitary, 17

unbounded,6

Ordinary differential operators

analyticity of semigroups generated by,

187-199

assignation of boundary conditions in

Co[0,oo), 135-137, in LP(0,oo), 137-138,

in C[0, 11, 143-144, in LP(0, /), 144

Notation and Subject Index

compactness of semigroups generated by,

198

formal adjoint, 129

formally self adjoint, 134

m-dissipative extensions in Co(- oo, oo),

129-134

m-dissipative extensions in LP(- oo, oo),

129-134

m-dissipative extensions in Co[0,oo)

Co,o[0,oo), 139-141

m-dissipative extensions in LP(0,oo),

139-141

m-dissipative extensions in C[0, /1,

Co, C,, C0 1, 143-145

m-dissipative extensions in LP(0, 1),

143-145

variational form, 134

Orthogonal of a set, 12

Outer normal vector, 204

Output (of a system), 479

P(d X) (resolution of the identity associated

with a normal operator), 18-19

P,,, 317

U(P), 485

(A; P), 3(A) (characteristic function of

an operator-valued distribution), 485

Paley-Wiener theorem, 105

Parabolic partial differential equations,

125-146, 187-266, 419-424

analyticity of solutions, 187-199, 255-263

boundary conditions, 135-136, 143-144,

204-214

compactness of solution operators, 263-264

conservation of L'-norm of nonnegative

solutions, 264

in spaces of continuous functions, 131-133,

141, 145, 244-245, 258-261

in L2 spaces, 131-133, 141, 145, 214-222

in LP spaces, 131-133, 141, 145, 236-243,

244-258,419-424

in one space variable, 125-146, 187-199,

419-423

in several space variables, 199-266,

423-424

nonnegative solutions, 263-264

time-dependent, 419-424

Partial differential operators (see Elliptic

operators, Symmetric hyperbolic

systems)

Partition of unity (subordinate to a cover),

205

Notation and Subject Index

Perturbations

of the Dirac operator, 296-297

of elliptic differential operators, 304-305

of essentially self-adjoint operators, 290

of infinitesimal generators, 338

of m-dissipative operators, 286-290

of the neutron transport operator, 279-280,

285-286

of operators in 6, 290-292

of operators in (2, 268-272

of operators in e+(1,0), 286-290

of the Schrodinger operator for I particle,

292-293, 297-304

of the Schrodinger operator for n particles,

293-296

of self-adjoint operators, 290, 301-302

of self-adjoint operators by symmetric

sesquilinear forms, 302

singular, 340-345

of stable families, 427

Perturbation problem, 338

Phillips adjoints (or #-adjoints), 77-80

of elliptic partial differential operators,

244-253

of m-dissipative operators, 125

of operators in e+, 77-80

of operators in (2+(1,0) 125

of ordinary differential operators, 133-134,

142, 146

of strongly continuous semigroups, 77-80

Poincare (asymptotic in the sense of), 71

Poisson formula (abstract), 100

Positive operators, 162

Positive part (of an element in a Banach

lattice), 160

Positive propagators, 162

Post formula (for inversion of Laplace

transforms), 92

Potentials

Coulomb for I particle, 293

Coulomb for n particles, 294

with inverse quadratic singularities,

300-304

Principle of fading memory, 481

Product integral, 425

Projection, 17

Propagators

isometric in t > 0, 154-156

isometric in - oo < t < oo, 121-122

isometric in Hilbert space, 159

positive, 162-164

of time-independent equations, 30

of time-dependent equations, 382

633

Quasilinear equations, 112-115

Quasi-reversibility methods, 344-345

r, r(A) (spectral radius of an operator), 6

R (X), R (A; A) (resolvent of an operator),

2, 7-8

p, p (A), (resolvent set of an operator), 2,

7-8

R (X), 91(A; P) (resolvent of an operator-

valued distribution), 485

r, r ( P) (resolvent set of an operator-valued

distribution), 485

r, r(O; A), r(X0), 186

R"', m-dimensional Euclidean space

R'+, 239

Real vs. complex spaces, 123-125

Reflexive spaces, 5, 125

Representation problem, 338

Resolution of the identity for a normal

operator, 18

Resolvent

of adjoint operator, 14

analytic continuation of, 8-9

of a bounded operator, 2

of an operator-valued distribution, 485

of an unbounded operator, 7

Resolvent equation

first, 8

second, 9

Resolvent set

of adjoint operator, 14

of a bounded operator, 2

of an operator-valued distribution, 485

of an unbounded operator, 7

Reverse logarithmic region, 494

Reversed Cauchy problem

for the heat equation in a square, 350-352

for abstract parabolic equations, 354-357

by logarithmic convexity methods, 375

Riesz potential operator of order - I, 311

Riesz representation theorem

for LF(Q; µ) spaces, 20

for C(K), Cj(K ), C0(K) spaces, 20-21

Riesz transforms, 311

Right (product) integral, 456

S(t), S(t; A), 30, 80, 272

S(t,s), 382

S(t; A) F, 307

SP(t), 188, 261

a, a(A) (spectrum of an operator), 2, 8

P) (spectrum of an operator-valued

distribution), 485

634

, (O; A), 186

E(K),

E,,(K), 21-22

E(-oo,oo), 133

236

E(9p), E+(q ), 175

E+(rp-), 180

E+(92, a), E+(9p - a). 180

S(R"'), S, 22

5'(E), 471

',(E), 472

S +(E ), 472

50'((X;

E)), 487

5o((X; E))n 6D ,((E; X))-', 493

So n GD I, Son

485

S' , S"(0, I), 204

204

Si", 204

Sato hyperfunctions, 505

Scattering theory, 109-110

Schrodinger equation

abstract, 60

conservation of norm of solutions, 40

for a free particle, 39-41, 504

for a particle, 292-293, 297-304

for n particles, 293-296

in L2(Rm), 39-401, 292-296, 297-304

in LP(R'"), 40-41, 504

perturbation theory of, 292-304

Self-adjoint operators, 17

Semigroups

analytic, 176-187

contraction, 118

discontinuous, 93-94

discrete, 329

generator of a discrete semigroup, 329

hyperfunction, 505

in the mildly well posed Cauchy problem,

502-503

infinitesimal generator of, 81, 505

in linear topological spaces, 102

isometric in t > 0, 154-156

isometric in - oo < t < oo, 121-122

isometric in Hilbert space, 159

of class Co, 94

of class (1, C1 ), 94

of class (1, A), 94

of class (0, C1 ), 94

of class (0, A), 94

of growth n, 94

positive, 162-164

special, 168

strongly continuous, 80-86

strongly measurable, 86

Notation and Subject Index

ultradistribution, 505

uniformly continuous, 83, 85

Semigroup equation(s), 63, 80

generalized, 505

Separable spaces, 15

Sine functional equation(s), 99

Singular differential equations, 110-111

Singular perturbations, 340-345

Sobolev imbedding theorems, 218-219, 234,

249

Sobolev spaces, 24-25

Solution of an abstract differential equation

explosive, 349

generalized, 30, 113

in t > 0, 29

inoo<t<oo,43

mild, 88, 453, 455

for time-dependent equations, 382, 387;

definition (A), 395; definition (B) for

homogeneous equation, 406; definition

(B) for inhomogeneous equation, 415;

definition (B'), 409; definition (C), 427;

definition (D), 436, 440; definition (E),

455

weak, 30, 89, 406, 415

Solution operator, see Propagator

Special semigroups, 168

Spectral measure, 17

Spectral radius of an operator, 6

Spectrum

of an operator, 2,

of a vector-valued distribution, 485

Stability constants C, w (of a stable family),

425

Stable families of operators, 425

State equation (of a system), 480

State space, 94

Stone's theorem, 91

Strong extension (of a symmetric hyperbolic

operator), 167

Strongly continuous (operator-valued) func-

tions, 3

Support

of a function, 22

of a vector-valued distribution, 464

Symmetric operators, 16

Symmetric hyperbolic systems

with constant coefficients in L2(R

43-45; in H `(R')° 45-46; in LP(Rm)°

49-50; LP and Co estimates for

solutions, 50-51

with variable coefficients in L2(R"

146-154, 159; in H'(R'")°, 306-316

Notation and Subject Index

with time-dependent coefficients in

L2(R"')`, 437, 449-450; in H'(R')',

450-451

Symmetric sesquilinear forms, 302

Systems, system theory, 94-97

causality, 479

continuity, 479

Green distribution, 480

Huygens' principle, 94-95, 507

initial state, 480

invertible, 480

memoryless, 481

Principle of fading memory for, 481

reversible, 95

state equation, 480

time-invariant, 95, 480

with finite memory, 481

9 (duality map), 120

Th (translation operator), 477

Test functions (spaces of)

6 (Q),6,22,461

5, 22, 470-71

bounded sets in' (d2), 463

bounded sets in 5, 471

convergence in 6D(Q), 461

convergence in 5, 470

Time-dependent

abstract differential equations, 381-460

abstract parabolic equations, 390-419

parabolic partial differential equations,

419-424

symmetric hyperbolic systems, 437-451

Total variation Ij

of a measure µ, 21

Trace, 219

Translation operator in 6 '(E), 477

Translation invariant operators in G1 +(E),

477

Translation semigroups, 74-76

Transmutation formulas, 105-108

Triangle inequality in a Banach lattice, 161

Trotter-Kato theorem, 337

Truncation at zero, 481

u* (element of dual space), 4

u+ (positive part of an element in a Banach

lattice), 160

u - (negative part of an element in a Banach

lattice), 160

(u*, u), u*(u), (u, u*), 4

635

(u*, u)P, 193

u V v (least upper bound of two elements

in a Banach lattice), 160

u A v (greatest lower bound of two elements

in a Banach lattice), 160

uI (absolute value or modulus of an element

in a Banach lattice), 160

U11 = u (convergence in an approximating

sequence of Banach spaces), 316

U®u, 462

U * V (convolution of two vector-valued

distributions), 468-470

Ultradistributions, 504

Ultradistribution semigroups, 505

Unbounded operators, 6

Uniform boundedness theorem, I

Uniformly bounded

cosine functions, 170

groups, 170

semigroups, 170

Uniform convergence on compacts of

t >_ t(e), 342

Uniformly elliptic partial differential

operators, 214

Uniqueness (of solutions of abstract

differential equations), 104-105

Unitary operators, 17

Vh (boundary patch), 205

V' (interior patch), 205

Vector-valued distributions, 461

abstract parabolic, 494

adjoint, 500

bounded convergence, 468

bounded sets in 6D'(2, E), 467; in 5'(E),

471

characteristic function, 485

convergence in 6D'(12; E), 463; in 5'(E),

471

convolution in 6D,, 468; in S+, 472; in

S.,476

convolution inverse, 480, 483

derivative, 462

fractional derivative, 477

Laplace transform in So(E), 473; of

distributions with compact support,

475

resolvent set, 485

resolvent, 485

spaces

of: 6'(12; E), 61'(E), 462; 61 o(E),

6D+(E), 6Do(Q; E), 464;

6;'(E), 6o(E),

E), 464;

636

S'(E), 471; 5,(E), 472 SL(E), 473

support,

464-465

Volterra's infinitesimal calculus, 425

W''P(2), 24

Wo °, 248

W (S7), 25

W2 P(12)p 235

Wave operators, 109

Weak convergence, 15-16

Weak extension (of a symmetric hyperbolic

operator), 167

Notation and Subject Index

Weak solutions (of abstract differential

equations), 30, 89, 406, 415

Weierstrass approximation theorem (by

semigroup methods), 85

Weierstrass semigroup, 116

s=(xl,...,x,,,_.1), 239

Y, 477

Yosida approximation

for time-independent equations, 92

for time-dependent equations, 458

Young's theorem, 23