Partially Specified Matrices and Operators: Classification, Completion, Applications

I. Main problems and motivation.- I.1 Eigenvalue completion problems and first examples.- I.2 Reduction by similarity.- I.3 Blocks.- I.4 Block similarity.- I.5 Special cases of block similarity.- I.6 Eigenvalue completion and restriction problems.- Notes.- II. Elementary operations on blocks.- II.1 Block-invariant subspaces.- II.2 Direct sums of blocks and decomposable blocks.- II.3 Indecomposable blocks.- II.4 Duality of blocks.- Notes.- III. Full length blocks.- III.1 Structure theorems for full length blocks.- III.2 Finite dimensional operator pencils.- III.3 Similarity of non-everywhere defined linear operators.- III.4 Dual sequences.- Notes.- IV. The eigenvalue completion problem for full length blocks.- IV.1 Main theorems.- IV.2 Reduction to a problem on matrix polynomials.- IV.3 A one column completion problem for matrix polynomials.- IV.4 Proof of the first main theorem.- IV.5 Some applications of the restriction problem.- IV.6 A matrix equation.- Notes.- V. Full width blocks.- V.1 Structure theorems for full width blocks.- V.2 Finite dimensional operator pencils.- V.3 Similarity of operators modulo a subspace.- V.4 Duality.- V.5 The eigenvalue completion problem and related problems.- V.6 A matrix equation.- Notes.- VI. Principal blocks.- VI.1 Structure theorem for principal blocks.- VI.2 The eigenvalue completion problem for principal blocks.- VI.3 The eigenvalue restriction problem for principal blocks.- Notes.- VII. General blocks.- VII.1 Block similarity invariants, completion and restriction problems.- VII.2 Structure theorems and canonical form.- VII.3 Proof of Proposition 2.2.- VII.4 Proof of Theorems 1.1 and 2.1.- VII.5 Finite dimensional operator pencils.- VII.6 Non-everywhere defined operators modulo a subspace.- VII.7 Duality of operator blocks.- VII.8 The eigenvalue completion problem.- Notes.- VIII. Off-diagonal blocks.- VIII.1 Structure theorems for off-diagonal blocks.- VIII.2 The eigenvalue completion and restriction problems.- Notes.- IX. Connections with linear systems.- IX.1 Linear input/output systems and transfer functions.- IX.2 Blocks and controllability.- IX.3 Blocks and observability.- IX.4 Minimal systems.- IX.5 Feedback and block similarity.- IX.6 Eigenvalue assignment and eigenvalue completion.- IX.7 Assignment of controllability indices and eigenvalue restriction.- IX.8 (A, B)-invariant subspaces.- IX.9 Output stabilization by state feedback.- IX.10 Output injection.- Notes.- X. Applications to matrix polynomials.- X.1 Preliminaries.- X.2 Matrix polynomials with prescribed zero structure.- X.3 Wiener-Hopf factorization and indices.- Notes.- XI. Applications to rational matrix functions.- XI.1 Preliminaries on pole pairs and null pairs.- XI.2 The one sided homogeneous interpolation problem.- XI.3 Homogeneous two sided interpolation.- XI.4 An auxiliary result on block similarity.- XI.5 Factorization indices for rational matrix functions.- Notes.- XII. Infinite dimensional operator blocks.- XII.1 Preliminaries.- XII.2 Main theorems about (P, I)-blocks.- XII.3 Main theorems for (I, Q)-blocks.- XII.4 Operator blocks on a separable Hilbert space.- XII.5 Spectral completion and assignment problems.- Notes.- XIII. Factorization of operator polynomials.- XIII.1 Preliminaries on null pairs and spectral triples.- XIII.2 Wiener-Hopf equivalence.- XIII.3 Wiener-Hopf factorization.- XIII.4 Wiener-Hopf factorization and strict equivalence.- XIII.5 The Fredholm case.- Notes.- XIV. Factorization of analytic operator functions.- XIV.1 Preliminaries on spectral triples.- XIV.2 Wiener-Hopf equivalence.- XIV.3 Wiener-Hopf factorization.- Notes.- XV. Eigenvalue completion problems for triangular matrices.- XV.1 U-similar and decomposed U-specified matrices.- XV.2 Invariants for U-similarity.- XV.3 Invariants and U-canonical form in the generic case.- XV.4 The diagonal of U-similar matrices.- XV.5 An eigenvalue completion problem.- XV.6 Applications.- Notes.- Append.- A.1 Root functions of regular analytic matrix functions.- A.2 Right Jordan pairs of regular analytic matrix functions.- A.3 Left Jordan pairs.- A.4 Jordan pairs and Laurent principal parts.- A.5 Global spectral data for regular analytic matrix functions.- Notes.- List of notations.