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Compression (functional analysis)

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563: 95: 349: 527: 138: 462: 423: 380: 288: 260: 240: 213: 604: 628: 48: 296: 597: 467: 623: 578: 104: 590: 538: 173: 435: 570: 395: 141: 401: 358: 8: 17: 273: 245: 225: 198: 157: 383: 36: 25: 574: 550:
P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982.
617: 430: 32: 216: 562: 470: 438: 404: 361: 299: 276: 248: 228: 201: 107: 51: 521: 456: 417: 374: 343: 282: 254: 234: 207: 132: 89: 148:. This is a natural way to obtain an operator on 615: 152:from an operator on the whole Hilbert space. If 529:, and we acquire the special definition above. 90:{\displaystyle P_{K}T\vert _{K}:K\rightarrow K} 598: 344:{\displaystyle T_{W}=V^{*}TV:W\rightarrow W} 66: 605: 591: 522:{\displaystyle V^{*}=I^{*}=P_{K}:H\to W} 616: 191:More generally, for a linear operator 557: 133:{\displaystyle P_{K}:H\rightarrow K} 13: 14: 640: 561: 513: 448: 335: 124: 81: 1: 544: 577:. You can help Knowledge by 7: 629:Mathematical analysis stubs 532: 425:is also self-adjoint. When 10: 645: 556: 164:, then the compression of 457:{\displaystyle I:W\to H} 398:, then the compression 573:–related article is a 523: 458: 419: 376: 345: 284: 256: 236: 209: 134: 91: 571:mathematical analysis 524: 459: 420: 418:{\displaystyle T_{W}} 396:self-adjoint operator 377: 375:{\displaystyle V^{*}} 346: 285: 257: 237: 210: 142:orthogonal projection 135: 92: 468: 436: 402: 359: 297: 274: 246: 226: 199: 105: 49: 624:Functional analysis 429:is replaced by the 195:on a Hilbert space 18:functional analysis 519: 454: 415: 372: 341: 280: 252: 232: 205: 158:invariant subspace 130: 87: 586: 585: 283:{\displaystyle W} 255:{\displaystyle H} 235:{\displaystyle W} 208:{\displaystyle H} 636: 607: 600: 593: 565: 558: 528: 526: 525: 520: 506: 505: 493: 492: 480: 479: 463: 461: 460: 455: 424: 422: 421: 416: 414: 413: 381: 379: 378: 373: 371: 370: 350: 348: 347: 342: 322: 321: 309: 308: 289: 287: 286: 281: 261: 259: 258: 253: 241: 239: 238: 233: 214: 212: 211: 206: 139: 137: 136: 131: 117: 116: 96: 94: 93: 88: 74: 73: 61: 60: 42:is the operator 644: 643: 639: 638: 637: 635: 634: 633: 614: 613: 612: 611: 554: 547: 535: 501: 497: 488: 484: 475: 471: 469: 466: 465: 437: 434: 433: 409: 405: 403: 400: 399: 366: 362: 360: 357: 356: 317: 313: 304: 300: 298: 295: 294: 275: 272: 271: 247: 244: 243: 227: 224: 223: 200: 197: 196: 112: 108: 106: 103: 102: 69: 65: 56: 52: 50: 47: 46: 26:linear operator 12: 11: 5: 642: 632: 631: 626: 610: 609: 602: 595: 587: 584: 583: 566: 552: 551: 546: 543: 542: 541: 534: 531: 518: 515: 512: 509: 504: 500: 496: 491: 487: 483: 478: 474: 453: 450: 447: 444: 441: 412: 408: 369: 365: 353: 352: 340: 337: 334: 331: 328: 325: 320: 316: 312: 307: 303: 279: 251: 231: 222:on a subspace 204: 129: 126: 123: 120: 115: 111: 99: 98: 86: 83: 80: 77: 72: 68: 64: 59: 55: 9: 6: 4: 3: 2: 641: 630: 627: 625: 622: 621: 619: 608: 603: 601: 596: 594: 589: 588: 582: 580: 576: 572: 567: 564: 560: 559: 555: 549: 548: 540: 537: 536: 530: 516: 510: 507: 502: 498: 494: 489: 485: 481: 476: 472: 451: 445: 442: 439: 432: 431:inclusion map 428: 410: 406: 397: 393: 389: 385: 367: 363: 338: 332: 329: 326: 323: 318: 314: 310: 305: 301: 293: 292: 291: 277: 269: 265: 262:, define the 249: 229: 221: 218: 202: 194: 189: 187: 183: 179: 175: 171: 167: 163: 159: 155: 151: 147: 143: 127: 121: 118: 113: 109: 84: 78: 75: 70: 62: 57: 53: 45: 44: 43: 41: 38: 34: 33:Hilbert space 30: 27: 23: 19: 579:expanding it 568: 553: 426: 391: 387: 354: 267: 263: 219: 192: 190: 185: 181: 177: 169: 165: 161: 153: 149: 145: 100: 39: 28: 21: 15: 264:compression 22:compression 618:Categories 545:References 174:restricted 514:→ 490:∗ 477:∗ 449:→ 368:∗ 336:→ 319:∗ 178:K→K 176:operator 125:→ 82:→ 539:Dilation 533:See also 217:isometry 180:sending 37:subspace 384:adjoint 382:is the 215:and an 172:is the 140:is the 355:where 156:is an 101:where 20:, the 569:This 394:is a 390:. If 144:onto 35:to a 31:on a 24:of a 575:stub 160:for 386:of 290:by 270:to 266:of 242:of 184:to 168:to 16:In 620:: 464:, 188:. 186:Tk 606:e 599:t 592:v 581:. 517:W 511:H 508:: 503:K 499:P 495:= 486:I 482:= 473:V 452:H 446:W 443:: 440:I 427:V 411:W 407:T 392:T 388:V 364:V 351:, 339:W 333:W 330:: 327:V 324:T 315:V 311:= 306:W 302:T 278:W 268:T 250:H 230:W 220:V 203:H 193:T 182:k 170:K 166:T 162:T 154:K 150:K 146:K 128:K 122:H 119:: 114:K 110:P 97:, 85:K 79:K 76:: 71:K 67:| 63:T 58:K 54:P 40:K 29:T

Index

functional analysis
linear operator
Hilbert space
subspace
orthogonal projection
invariant subspace
restricted
isometry
adjoint
self-adjoint operator
inclusion map
Dilation
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mathematical analysis
stub
expanding it
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Categories
Functional analysis
Mathematical analysis stubs

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