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This is the Habilitation Thesis manuscript presented at Besan\c{c}on on January 5, focusing on Matrix Analysis, Matrix Inequalities and Matrix Decompositions. There are also some topics in (Hilbert space) Operator Theory. The text should be of interest for a large audience of researchers and students in pure and applied mathematics. We may divide it into five parts: 1) Chapter 1 is an introductory chapter, some results from the period 1999-2010 are given, and a few conjectures are proposed. 2) Chapters 2-4 deal with matrix inequalities, Chapter 2 is concerned with norm inequalities and logmajorization and Chapters 3-4 with functional calculus and a unitary orbit technique that I started to develop in 2003. 3) Chapter 5 is a time-break in infinite dimensional Hilbert space operators, the essential numerical range plays a key role. 4) Chapters 6-8 establish several decompositions for partitioned matrices, especially for positive block matrices. Some norm inequalities involving the numerical range are derived. 5) Chapter 9 may be of special interest for students : a proof of the Spectral Theorem for bounded operators is derived from the matrix case., Comment: 144 pages. This is the Habilitation Thesis manuscript presented at Besan\c{c}on on January 5. Special thanks are due to my colleagues and friends kindly acting as rewievers, Frank Hansen, Hideki Kosaki, and Fedor Sukochev

We establish a sharp inequality between the blocks of positive partitioned matrices and conjecture a triangle type inequality for contractions: Given three contactions A,B,C, we conjecture that the constant c=3/4 is sharp in the triangle inequality |A+B+C| < cI + |A| +|B| + |C|., Comment: Final version, to appear in LAA, 13 pages

We study an elementary inequality supporting the classical Hermite-Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such new Schatten p-norm estimates and new majorization, Comment: To appear in Canadian Math. Bulletin

We prove some eigenvalue inequalities for positive semidefinite matrices partitioned into four blocks. The inradius of the numerical range of the off-diagonal block contributes to these estimates. Some related norm inequalities are given and a conjecture is proposed., Comment: To appear in International J. Math

We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities.

For a positive linear map F and a normal matrix N, we show that |F(N)| is bounded by some simple linear combinations in the unitary orbit of F(|N|). Several elegant sharp inequalities are derived, especially for the Schur product.

We extend some inequalities for normal matrices and positive linear maps related to the Russo-Dye theorem. The results cover the case of some positive linear maps on a von Neumann algebra mapping any nonzero operator to an unbounded operator.

A refinement of a trace inequality of McCarthy establishing the uniform convexity of the Schatten $p$-classes for p>2 is proved

We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range of the off-diagonal block X, especially the distance from 0 to W(X).

We revisit a classical result, the Russo-Dye Theorem, stating that every positive linear map attains its norm at the identity.

We study various convex functions on $R^n$ associated with positive definite matrices. This yiels some exotic Holder matrix inequalities., Comment: 6 pages

Several matrix/operator inequalies are given. Most of them are unexpected extensions of the Araki Log-majorization theorem, obtained thanks to a new log-majorization for positive linear maps and normal operators (Theorem 2.9). The main idea and technical tool is a two variables log-convex norm functional (Theorem 1.2)., Comment: Final version, to appear in International J. Math: Two corollaries on Schur products have been added at the end of Section 2

We employ the pinching theorem, ensuring that some operators A admit any sequence of contractions as an operator diagonal of A, to deduce/improve two recent theorems of Kennedy-Skoufranis and Loreaux-Weiss for conditional expectations onto a masa in the algebra of operators on a Hilbert space. We also get a few results for sums in a unitary orbit.

As the reversed version of usual symmetric norms, we introduce the notion of symmetric anti-norms $\|\cdot\|_!$ defined on the positive operators affiliated with a finite von Neumann algebra with a finite normal trace. Related to symmetric anti-norms, we develop majorization theory and superadditivity inequalities of the form $\|\psi(A+B)\|_!\ge\|\psi(A)\|_!+\|\psi(B)\|_!$ for a wide class of functions $\psi$., Comment: 27 pages, largely revised from the previous version, to appear in Publ. Res. Inst. Math. Sci