Subnormality of Powers of Multivariable Weighted Shifts

<div>Weight diagram of the 2-variable weighted shift <svg height="9.65132pt" id="M200" style="vertical-align:-0.6370001pt" version="1.1" viewbox="-0.0498162 -9.01432 44.9177 9.65132" width="44.9177pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M616 481C611 545 606 633 605 675H580C563 656 549 650 523 650H121C92 650 79 653 64 675H39C37 624 31 551 27 478H64C77 528 89 562 106 579C120 596 138 606 221 606H255V133C255 48 245 42 161 35V0H487V35C399 42 389 48 389 133V606H434C493 606 518 601 536 583S568 531 580 479L616 481Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.996,0)"><path d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.492,0)"><path d="M896 675C863 652 826 634 783 634C683 634 549 687 439 687C259 687 128 601 128 473C128 383 196 329 279 329C392 329 469 405 496 541H476C440 428 385 359 286 359C224 359 169 400 169 480C169 566 257 641 382 641C475 641 622 584 731 584C796 584 862 620 907 657L896 675ZM658 553C579 484 521 424 454 277C364 81 295 16 184 16C120 16 82 44 78 79H80C87 68 100 67 108 67C136 67 153 89 153 116S131 165 104 165C69 165 43 139 43 91C43 31 110 -15 191 -15C263 -15 321 6 378 45C463 101 526 197 557 304C588 413 618 487 671 539L658 553Z"></path></g><g transform="matrix(.013,0,0,-0.013,34.348,0)"><path d="M449 645C392 682 336 687 293 687C136 687 37 594 37 499C37 397 122 342 207 329C176 273 159 217 159 158C159 48 247 -15 356 -15C417 -15 478 3 530 39S626 128 626 187C626 256 580 290 508 290C402 290 311 187 311 65L333 63C344 171 423 247 491 247C528 247 562 225 562 173C562 97 470 15 360 15C284 15 243 61 243 125C243 176 255 240 312 324C552 324 781 473 781 594C781 642 748 669 685 669C547 669 336 527 229 361C151 369 78 413 78 501C78 604 200 657 301 657C336 657 381 657 437 626L449 645ZM331 355C443 513 593 632 690 632C723 632 740 619 740 592C740 534 671 464 576 416C501 377 411 354 332 354L331 355Z"></path></g></svg> and weight diagram of the 2-variable weighted shift <span class="nowrap"><svg height="9.65132pt" id="M201" style="vertical-align:-0.6370001pt" version="1.1" viewbox="-0.0498162 -9.01432 44.5653 9.65132" width="44.5653pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M616 481C611 545 606 633 605 675H580C563 656 549 650 523 650H121C92 650 79 653 64 675H39C37 624 31 551 27 478H64C77 528 89 562 106 579C120 596 138 606 221 606H255V133C255 48 245 42 161 35V0H487V35C399 42 389 48 389 133V606H434C493 606 518 601 536 583S568 531 580 479L616 481Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.996,0)"><path d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.492,0)"><path d="M810 587C770 574 761 568 725 543C654 630 540 687 408 687C217 687 69 577 69 430C69 319 160 247 276 247C369 247 511 319 511 495C511 510 512 523 509 538H490C490 354 393 277 274 277C174 277 110 344 110 422C110 559 219 655 415 655C524 655 636 602 695 519C628 463 572 392 516 303C433 170 377 97 313 57C252 89 199 109 143 109C79 109 36 87 36 54C36 15 82 -9 165 -9C222 -9 276 0 326 18C373 -1 423 -15 482 -15C645 -15 804 133 804 324C804 393 783 458 747 514C776 546 781 551 818 567L810 587ZM756 336C756 135 623 16 478 16C439 16 402 24 367 36C386 45 401 56 420 68C505 127 569 206 616 302C647 362 679 427 719 480C744 435 756 387 756 336ZM276 39C243 26 207 21 164 21C106 21 77 32 77 54C77 75 125 79 144 79C191 79 227 66 276 39Z"></path></g><g transform="matrix(.013,0,0,-0.013,33.997,0)"><path d="M449 645C392 682 336 687 293 687C136 687 37 594 37 499C37 397 122 342 207 329C176 273 159 217 159 158C159 48 247 -15 356 -15C417 -15 478 3 530 39S626 128 626 187C626 256 580 290 508 290C402 290 311 187 311 65L333 63C344 171 423 247 491 247C528 247 562 225 562 173C562 97 470 15 360 15C284 15 243 61 243 125C243 176 255 240 312 324C552 324 781 473 781 594C781 642 748 669 685 669C547 669 336 527 229 361C151 369 78 413 78 501C78 604 200 657 301 657C336 657 381 657 437 626L449 645ZM331 355C443 513 593 632 690 632C723 632 740 619 740 592C740 534 671 464 576 416C501 377 411 354 332 354L331 355Z"></path></g></svg>,</span> respectively.</div>

Journal of Function Spaces

Subnormality of Powers of Multivariable Weighted Shifts

Figure 1