Linear Operators: General theory |
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Page 45
... matrix ( a ;; ) . If T is a linear operator , then it can be shown that the determinants of the matrices of T in terms of any two bases are equal , and so we may and shall call their common value the determinant of T , denoted by det ...
... matrix ( a ;; ) . If T is a linear operator , then it can be shown that the determinants of the matrices of T in terms of any two bases are equal , and so we may and shall call their common value the determinant of T , denoted by det ...
Page 561
... matrices A in Em . Since e - t1t = I , the columns of the matrix et form a set of m linearly independent solutions of the equation dy / dt = Ay . ( d ) If T is such that r ( 2 ) = 1 for λ e σ ( T ) , Theorem 8 yields = k ƒ ( T ) = Σƒ ...
... matrices A in Em . Since e - t1t = I , the columns of the matrix et form a set of m linearly independent solutions of the equation dy / dt = Ay . ( d ) If T is such that r ( 2 ) = 1 for λ e σ ( T ) , Theorem 8 yields = k ƒ ( T ) = Σƒ ...
Page 564
... matrix differential equation dy / dt = A ( t ) Y where a solution is a nxn complex valued matrix Y ( t ) , differentiable and satisfying the differential equation for each te I. Show that for each to e I and each complex ( constant ) matrix ...
... matrix differential equation dy / dt = A ( t ) Y where a solution is a nxn complex valued matrix Y ( t ) , differentiable and satisfying the differential equation for each te I. Show that for each to e I and each complex ( constant ) matrix ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ