Index
absolute value, 12
adjoint, 57
Ando-Okubo theorem, 85
antisymmetric tensor power, 93
antisymmetric tensor product, 44
approximate identity, 169
arithmetic-geometric mean inequality, 102, 198
for matrices, 160, 198
refinement of, 102, 132
arithmetic-logarithmic-geometric mean in-equality
for matrices, 163
for numbers, 163
Arveson’s extension theorem, 71
and Hahn-Banach theorem, 95
averaging operation, 38
Banach Alaoglu theorem, 166
Berger’s theorem, 98
Bernstein-Widder theorem, 194
Bessis-Moussa-Villani conjecture, 176
binomial means, 132
Bochner’s theorem, 168, 169, 189
and Stone’s theorem, 194
generalisations, 193
Bruhat-Tits space, 228
Cartesian decomposition, 39
CAT(0) inequalities, 234
Cauchy’s interlacing theorem, 7
Cauchy matrix, 3, 8, 161
determinant of, 30
infinitely divisible, 24
centre of mass, 216, 217
and matrix inversion, 232
and monotonicity, 217
E. Cartan’s theorem, 233
equation for, 220
of commuting matrices, 220
centroid, 216
channel errors, 96
characteristic function, 173
Choi’s inequality, 41
Choi’s theorem, 66
Cholesky decomposition, 2
circulant matrix, 48
comparison triangle, 227
completely bounded map, 99
completely monotone function, 194
completely positive map, 65, 95
as quantum channel, 96
dilation to representation, 68
extension of, 71
in noncommutative probability, 99
nonlinear, 92, 99
representation, 66
theorem of Choi and Kraus, 66
completion problem, 76
concavity
of matrix powers, 112
of matrix square root, 24
of tensor powers, 114
conditionally positive definite, 180
congruence invariance, 102
congruence transformation, 105, 201
as an isometry, 202
congruent matrices, 5
contraction, 13
convex, 18
convex map, 18
convex set, 212
convex hull, 212
of three points, 216
convexity
in differentiable manifolds, 233
joint, 19
of matrix inverse, 18
of matrix powers, 21, 23, 24, 113
of matrix square, 19
of metric, 229
of relative entropy, 119
of Riemannian metric, 209
of tensor products, 113
convolution, 149
covariance, 74
between functions, 74
between operators, 74
Daleckii-Krein formula, 60, 233
decimated Hamiltonian, 32
decimation map, 32
decomposable operator, 120
derivative, 44, 60
and Schur product, 60
norm of, 44, 45
of exponential, 225
Descartes rule of signs, 7, 30
differentiable manifold, 201
dilation theory, 31
entropy
classical, 114
of tensor products, 120
quantum, 115
relative, 116
skew, 116
strong subadditivity, 124
subadditivity, 121
theorem of Lieb and Ruskai, 124
entrywise positive, 24
exponential map
continuity of, 210
derivative of, 202, 225
metric increasing, 204
exponential metric increasing property
(EMI), 203
generalised, 223
exterior power, 44
exterior product, 44
Feshbach map, 32
Fiedler’s inequality, 19
Finsler manifolds, 227
first divided difference, 60, 154
Fourier’s law, 135
Fourier-Stieltjes coefficients, 4
Fourier-Stieltjes transform, 145
Fourier transform, 145, 169, 172
Frobenius inequality, 94
Furuta’s inequality, 126
geodesic, 202
and order, 230
existence of, 205
natural parametrisation of, 205
geometric mean, 105, 132, 136
and matrix inversion, 232
and Riccati equation, 106
as distance minimiser, 214
as geodesic midpoint, 206
as metric midpoint, 223
continuity of, 212
of 2 × 2 matrices, 111
of three matrices, 215, 222
Golden-Thompson inequality, 224
Grassmann power, 44
Grassmann product, 44
Haagerup’s theorem, 18, 79, 97
analogue for numerical radius, 85
Hahn-Banach theorem, 47, 51, 63
Hankel matrix, 194
harmonic mean, 103
and Schur complement, 103
concavity of, 104
Heinz means, 131, 139
of matrices, 229
Helly’s selection principle, 166, 169
Herglotz’ theorem, 4, 167, 175
Hilbert’s inequality, 30
Hilbert’s theorem, 96
Hilbert matrix, 30
inertia, 5
and Schur complement, 31
complete invariant for congruence, 5
infinitely divisible
distribution functions, 197
function, 184
matrix, 24
infinitesimal exponential metric increasing property (IEMI), 202
generalised, 223
integral representation, 21
Jensen’s inequality, 62
jointly concave map, 20
jointly convex map, 19
Kadison’s inequality, 39, 53, 54, 74
Choi’s generalisation, 40
Kantorovich inequality, 56
Kirchhoff’s laws, 103
Klein’s inequality, 118
Kraus’ theorem, 66
Krein extension theorem, 47, 51, 78
Ky Fan norm, 58
Laplace transform, 194
Lieb’s concavity theorem, 117
Ando’s proof, 113
Lieb-Ruskai theorem, 124
Lieb class, 87
linear map
adjoint, 57
doubly stochastic, 57
positive, 36
unital, 36
Loewner’s theorem, 196
Loewner matrix, 154
infinite divisibility of, 196
positivity of, 157
logarithmic mean, 135, 162, 229
and Riemannian geometry, 225
in heat flow, 136
of matrices, 229
logarithmic mean area, 136
logarithm of matrices
concave, 113
monotone, 113, 224
Lyapunov equation, 9, 43, 57, 130
discrete time, 10
perturbed, 44
solution, 10
m-positive, 65
and completely positive, 71
matrix
circulant, 48
conditionally positive definite, 180, 182
entrywise positive, 24
Hankel, 194
infinitely divisible, 24, 153
partially defined, 77
Pascal, 182
positive definite, 1
positively stable, 9
positive semidefinite, 1
stochastic, 55
Toeplitz, 4, 194
matrix convex function, 61
matrix mean, 102
arithmetic, 102
geometric, 102, 105, 108
harmonic, 102, 103
Kubo and Ando, 137
properties, 102
matrix means
and convexity, 111
and monotonicity, 111
and positive linear maps, 107
matrix monotone function, 60
derivative of, 61
matrix monotone of order n, 60
matrix units, 66
mean, 101
arithmetic, 101
binomial, 132
geometric, 101
harmonic, 101
Heinz, 131
logarithmic, 101
of matrices, 102
power, 132
means
domination between, 180
order between, 180
metric projection, 212
minimax principle, 5
Minkowski determinant inequality, 114
monotone map, 18
monotonicity of matrix powers, 22, 112, 125, 129
Naimark’s theorem, 95
noise operators, 96
nonnegative curvature, 227
nonpositive curvature, 226
metric space of, 227
norm
Hilbert-Schmidt, 57, 162
Ky Fan, 57
of positive linear map, 42
of Schur product, 16, 43, 59, 79, 80, 90, 91
operator, 12
Schatten, 57
trace, 57
unitarily invariant, 57
unitary invariance of, 12
numerical radius, 81
Ando’s theorem, 82
power inequality, 85
numerical range, 81
off-diagonal part, 90
norm of, 90
operator-valued measures, 94
operator convex function, 61
operator monotone, 33, 61
and Loewner matrix, 156
square root, 9
operator space
abstract, 64
concrete, 64
operator system, 47
completely positive map on, 70
Pólya’s theorem, 151
parallelogram law, 207
Pascal matrix infinitely divisible, 183
pattern, 77
Pick functions, 189
pinching, 37, 57
as average of unitary conjugates, 88
as noncommutative convex combination, 87
doubly stochastic, 57
norm of, 87
reduces norm, 43
polar decomposition, 12
positive completion, 77
positive completion problem, 77
positive definite
function, 142
kernel, 144
matrix, 1
sequence, 4, 141, 175
positive definite function, 142
and operator monotonicity, 157
applications, 191
group representations, 191
positive linear functional, 37, 46
extension of, 47
norm of, 51
positive linear map, 36, 46
2-positive behaviour of, 72, 96
and logarithm, 131
and logarithmic mean, 231
and matrix powers, 52, 53
and means, 107
as noncommutative expectation, 38
examples, 37
norm of, 42, 43, 47, 50, 52
preserves adjoints, 39, 50
positive matrices
as a Bruhat-Tits space, 228
as a Finsler manifold, 227
as a manifold of nonpositive curvature, 207
as a Riemannian manifold, 201
characterizations, 1
existence of geodesics, 205
Hadamard product of, 8
Riemannian metric on, 201, 202, 206
Schur product of, 8
square root of, 2
positive part, 12
positive type, 194
power inequality, 91
power means, 132
projection-valued measure, 94, 175
quantum channels, 96
relative entropy
and partial trace, 123
classical, 118
convexity of, 119
quantum, 118
representation, 35, 36
characterisation, 36
Riccati equation, 11, 31
positive solution, 11
Riemannian metric, 201, 206
and triangles, 216
completeness, 210
convexity of, 209
projection onto convex set, 212
Riemannian symmetric space, 226
Riesz representation theorem, 38, 166, 169
Russo-Dye theorem, 41, 43, 47, 63
Schatten p-norm, 58
Schur complement, 20, 32, 103
and determinant, 21
and harmonic mean, 103
and inertia, 31
concavity of, 21
in quantum mechanics, 32
Schur multiplier, 78
Schur product, 7, 19, 32, 57, 98
m-fold, 24
and derivative, 60
and inequalities for matrix means, 161
norm of, 16, 43, 59, 79, 80
Schwarz inequality, 73, 85, 97
operator versions, 73, 85
Segal’s inequality, 224
semiparallelogram law, 207, 208, 223, 233
in a metric space, 227
simultaneously congruent, 23
simultaneously diagonalizable, 23
singular value decomposition (SVD), 12
singular values, 12
skew entropy, 116
spectral theorem, 175
square root
concavity of, 24
of a positive matrix, 2
operator monotone, 9
with positive eigenvalues, 109, 130
Stein equation, 10, 44
perturbed, 44
solution, 11
Stinespring’s dilation theorem, 68, 94, 124
and Naimark’s theorem, 95
Stone’s theorem and Bochner’s theorem, 193
strictly convex, 210
Sylvester’s law of inertia, 5, 30
Sylvester equation, 130
t-power mean, 233
tangent space, 201
tensor powers, 93
convexity, 113
tensor product
in quantum information theory, 139
in statistical mechanics, 139
tent function, 149
Toeplitz matrices, 194
trace norm, 43
triangle inequality, 27
stronger, 27
triangular truncation norm of, 98
unitarily equivalent, 5
unitarily invariant, 57
unitarily invariant norms
and geodesics, 223
properties, 58
unitary dilation, 42
unitary invariance, 12
variance, 54, 74
inequality, 54
of a function, 54
of an operator, 54
variance-covariance inequality, 74
weak∗ topology, 166
Wielandt’s inequality, 86